What's ground? If I connect the shield of my coax (which is grounded outside) to the antenna input of my R8, I hear lots of junk, indicating that there is an RF voltage difference between the coax shield and the R8 chassis. Last night this measured about S5.5, which is about -93 dBm (preamp off, 6KHz bandwidth). That's a lot of noise: it was 18 dB above my antenna's "noise floor", and 26 dB above the receiver's noise floor.
This sort of disagreement about ground potential is characteristic of electrically noisy environments. The receiver will, of course, respond to any voltage input that differs from its chassis ground. The antenna, on the other hand, is in a very different environment, and will have its own idea of what ground potential is. If you want to avoid noise pickup, you need to deliver a signal, referenced at the antenna to whatever its ground potential is, in such a way that when it arrives at the receiver, the reference potential is now the receiver's chassis potential.

Coaxial cable represents one way to do this. Coax has two key properties:

1. The voltage between the inner conductor and the shield depends only on the state of the electromagnetic field within the shield.

2. The shield prevents the external electromagnetic field from influencing the internal electromagnetic field (but watch out at the ends of the cable!).

So, it's easy, right? Run coax from the antenna to the receiver. Ground at the antenna end will be whatever the antenna thinks it is, while ground at the receiver end will be whatever the receiver thinks it is. The antenna will produce the appropriate voltage difference at the input side, and the receiver will see that voltage difference uncontaminated by external fields, according to the properties given above.

Unfortunately, it doesn't quite work that way. It's all true as far as it goes, but it neglects the fact that the coax can also guide noise from your house to your antenna, where it can couple back into the cable and into your receiver. To see how this works, let me first describe how this noise gets around.
The noise I'm talking about here is more properly called "broadband electromagnetic interference" (EMI). It's made by computers, lamp dimmers, televisions, motors and other modern gadgets. I have all these things. In many cases, I can't get them turned off, because it would provoke intrafamilal rebellion. However, even when I turn them off, the noise in the house doesn't go down very much, because my neighbors all have them too. In any case, one of the worst offenders is my computer, which is such a handy radio companion I'm not about to turn it off.
Some of this noise is radiated, but the more troublesome component of this is conducted noise that follows utility wires. Any sort of cable supports a "common mode" of electromagnetic energy transport in which all of the conductors in the cable are at the some potential, but that potential differs from the potential of other nearby conductors ("ground"). The noise sources of concern generate common mode waves on power, telephone, and CATV cables which then distribute these waves around your neighborhood. They also generate "differential" mode waves, but simple filters can block these so they aren't normally a problem.

So, let's say you have a longwire antenna attached to a coaxial cable through an MLB ("Magnetic Longwire Balun" [sic]). Suppose your next door neighbor turns on a dimmer switch. The resulting RF interference travels out his power lines, in through yours, through your receiver's power cord to its chassis, and out your coaxial cable to your MLB. Now on coax, a common mode wave is associated with a current on the shield only, while the mode we want the signal to be in, the "differential" mode, has equal but opposite currents flowing on shield and inner conductor. The MLB works by coupling energy from a current flowing between the antenna wire and the coax shield into into the differential mode. But wait a second: the current from the antenna flows on the coax shield just like the common mode current does.
Does this mean that the antenna mode is contaminated with the noise from your neighbor's dimmer?
The answer is a resounding, and unpleasant, yes! The way wire receiving antennas work is by first moving energy from free space into a common mode moving along the antenna wire, and then picking some of that off and coupling it into a mode on the feedline. In this case, the common mode current moving along the antenna wire flows into the common mode of the coax, and vice versa. The coax is not just feedline: it's an intimate part of the antenna! Furthermore, as we've seen, it's connected back through your electrical wiring to your neighbor's dimmer switch. You have a circuitous but electrically direct connection to this infernal noise source. No wonder it's such a nuisance!

The solution is to somehow isolate the antenna from the common mode currents on the feedline. One common way to do this is with a balanced "dipole" antenna. Instead of connecting the feedline to the wire at the end, connect it to the middle. Now the antenna current can flow from one side of the antenna to the other, without having to involve the coax shield. Unfortunately, removing the necessity of having the coax be part of the antenna doesn't automatically isolate it: a coax-fed dipole is often only slightly quieter than an end-fed longwire. A "balun", a device which blocks common mode currents from the feedline, is often employed. This can improve the situation considerably. Note that this is not the same device as the miscalled "Magnetic Longwire Balun".
Another way is to ground the coaxial shield, "short circuiting" the common mode. Antenna currents flow into such a ground freely, in principle not interacting with noise currents. The best ground for such a purpose will be a earth ground near the antenna and far from utility lines.
Still another way is to block common mode waves by burying the cable. Soil is a very effective absorber of RF energy at close range.

Unfortunately, none of these methods is generally adequate by itself in the toughest cases. Baluns are not perfectly effective at blocking common mode currents. Even the best balun can be partially defeated if there's any other unsymmetrical coupling between the antenna and feedline. Such coupling can occur if the feedline doesn't come away from the antenna at a right angle. Grounds are not perfect either. Cable burial generally lets some energy leak through. A combination of methods is usually required, both encouraging the common mode currents to take harmless paths (grounding) and blocking them from the harmful paths (baluns and/or burial).
The required isolation to reach the true reception potential of the site can be large. According to the measurements I quoted above, for my site the antenna noise floor is 18 dB below the conducted noise level at 10 MHz. 18 dB of isolation would thus make the levels equal, but we want to do better than that: we want the pickup of common mode EMI to be insignificant, at least 5 dB down from the antenna's floor. In my location the situation gets worse at higher frequencies as the natural noise level drops and therefore I become more sensitive: even 30 dB of isolation isn't enough to completely silence the common mode noise (but 36 dB is enough, except at my computer's CPU clock frequency of 25 MHz).

Getting rid of the conducted noise can make a huge difference in the number and kinds of stations you can pick up: the 18 dB difference between the conducted and natural noise levels in the case above corresponds to the power difference between a 300 kW major world broadcaster and a modest 5 kW regional station.
The method I use is to ground the cable shield at two ground stakes and bury the cable in between. The scheme of alternating blocking methods with grounds will generally be the most effective. The ground stake near the house provides a place for the common mode noise current to go, far from the antenna where it cannot couple significantly. The ground stake at the base of my inverted-L antenna provides a place for the antenna current to flow, at a true ground potential relative to the antenna potential. The buried coax between these two points blocks noise currents.

There has been some discussion of grounding problems on this and related echos. I believe it has been mentioned that electrical codes require that all grounds be tied together with heavy guage wire.

I'm no expert on electrical codes, and codes differ in different countries. However, I believe that any such requirement must refer only to grounds used for safety in an electric power distribution system: I do not believe this applies to RF grounds.
Remember that proper grounding practice for electrical wiring has very little to do with RF grounding. The purpose of an electrical ground is to be at a safe potential (a few volts) relative to non-electrical grounded objects like plumbing. At an operating frequency of 50/60 Hz, it needs to have a low enough impedance (a fraction of an ohm) that in case of a short circuit a fuse or breaker will blow immediately.
At RF such low impedances are essentially impossible: even a few centimeters of thick wire is likely to exhibit an inductive impedance in the ohm range at 10 MHz (depends sensitively on the locations and connections of nearby conductors). Actual ground connections to real soil may exhibit resistive impedances in the tens of ohms. Despite this, a quiet RF ground needs to be within a fraction of a microvolt of the potential of the surrounding soil. This is difficult, and that's why a single ground is often not enough.

A little experimentation with my radio showed that the chassis was directly connected to the third (grounding) prong of the wall plug. I am concerned that by connecting my receiver to an outside ground I am creating a ground loop that involves my house wiring. Can you comment on this?

Yes, you have a "ground loop". It's harmless. In case of a nearby lightning strike it may actually save your receiver. My R8 isn't grounded like that, so I had to take steps to prevent the coax ground potential from getting wildly out of kilter with the line potential and arcing through the power supply. I'm using a surge supressor designed to protect video equipment: it has both AC outlets and feedthroughs with varistor or gas tube clamps to keep the various relative voltages in check. Of course the best lightning protection is to disconnect the receiver, but I'm a bit absent minded so I need a backup.

This may seem like a trivial point but I recently discovered that the main ground from the electrical service panel in my house was attached to a water pipe which had been painted over. I stripped the paint from the pipe and re-attached the grounding clamp and I noticed a reduction in noise from my receiver.

Not trivial. Not only did you improve reception, but your wiring is safer for having a good ground.
I suspect part of the reason I see so much noise from neighbors' appliances on my electric lines may be that my house's main ground wire is quite long. The electrical service comes in at the south corner of the house (which is where the breaker box is), while the water (to which the ground wire is clamped) enters at the east corner. All perfectly up to code and okay at 60 Hz, but lousy at RF: if it was shorter, presumably more of the noise current would want to go that way, and stay away from my receiver.

I am also a little confused by what constitues an adequate ground. I have read that a conducting stake driven into the ground will divert lightning and provides for electrical safety but that RF grounding systems have to be a lot more complex with multiple radials with lengths related to the frequencies of interest. Is this true?

Depends on what you're doing. If you're trying to get maximum signal transfer with a short loaded (resonant) vertical antenna with a radiation resistance of, say, 10 ohms, 20 ohms of ground resistance is going to be a big deal. If you're transmitting 50 kW, your ground resistance had better be *really* tiny or things are going to smoke, melt or arc.
On the other hand, a ground with a resistance of 20 ohms is going to be fairly effective at grounding a cable with a common mode characteristic impedance of a few hundred ohms (the characteristic impedance printed on the cable is for the differential mode; the common mode characteristic impedance depends somewhat on the distance of the cable from other conductors, but is usually in the range of hundreds of ohms). Of course, if it was lower a single ground might do the whole job (but watch out for mutual inductance coupling separate conductors as they approach your single ground).
In addition, a ground with a resistance of 20 ohms is fine for an unbalanced antenna fed with a high impedance transformer to supress resonance. Such a nonresonant antenna isn't particularly efficient, but high efficiency is not required for good reception at HF and below (not true for VHF and especially microwave frequencies).

Much antenna lore comes from folks with transmitters who, armed with the "reciprocity" principle, assume that reception is the same problem. The reciprocity principle says that an antenna's transmission and reception properties are closely related: it's good physics, but it ignores the fact that the virtues required of a transmitting and receiving antenna are somewhat different. Inefficiency in a transmitting antenna has a direct, proportional effect on the received signal to noise ratio. On the other hand, moderate inefficiency in an HF receiving antenna usually has a negligible effect on the final result. A few picowatts of excess noise on a transmitting antenna has no effect on its function, but is a big deal if you're receiving (of course, one might not want to have transmitter power going out via unintended paths like utility lines: this is indeed the "reciprocal" of the conducted noise problem, and has similar solutions).