This answer discusses contact binaries, which I did not even know existed. I can understand how they could exist for a short time (maybe) as gravitational waves carry off energy that causes the stars' orbits to slowly decay, but how can they persist? My main concern is that surely the stars orbit within the Roche limit of each other (or at least the smaller one orbits within the Roche limit of the larger one). Has there been any good theoretical work on their stability?
I agree with Walter, they don't. However, in addition to the common envelope drag and mass exchange a very important feature of their evolution is the loss of angular momentum through a magnetised wind. The loss of angular momentum from the orbit also leads to orbital shrinkage and closer contact, until presumabaly at some point they truly merge.
This mechanism is thought to be one of the main processes that lead to the blue straggler phenomenon - apparently over-luminous main sequence stars seen in older clusters.
You can try to observe this orbital shrinkage, but it takes place slowly enough, and their are other confusing pseudo-periodic phenomena that also lead to changes (up and down) in orbital period (e.g. the Applegate mechanism), so that it is very difficult to pin down the orbital shrinkage rate, or even verfy experimentally that it is happening.
An alternative approach is to estimate the age distributions of contact binaries by looking at their kinematics compared to the expected time for short-period binaries to evolve into a contact state. These numbers are still highly debatable and uncertain; Eker et al. (2008) use this approach to estimate a timescale of 1.6 Gyr for the contact stage.
Short answer: They don't.
Contact binaries are a possible evolutionary phase of close binary systems, where both stars fill and/or overflow their Roche limit, i.e. 'kiss' each other at the L$_1$ point. As a consequence, they lose matter to a common envelope. This exerts a drag onto the stars and leads to further shrinking of the binary, which eventually merges. AFAIK, this process is mainly hypothetical (though sensible), i.e. no direct observational evidence exits, and its details are still rather unclear.