Over at RPGnet, they have a new entry in a column called Duets that gives advice on running 1-on-1 RPGs (ie: just a player and a GM). Having done this on several occasions in the past, it can definitely be a rewarding experience, though I prefer at least 2 or 3 players, generally.
The obvious advantage to duets is that they are easier to schedule, especially, as most duets involve couples, siblings, roommates, and the like who can game as much as they want with little hassle over scheduling because they live together. I have run a lot of group campaigns over the years, but duets with my wife have allowed us to log an insane amount of time roleplaying over the last 15 years. This is simply because duets give us the chance to roleplay most evenings. Another advantage is that duets are more intense due to their quicker pace, greater immersion, and increased roleplaying. Duets also allow you to more thoroughly explore a theme or story arc as everything is focused on one player and there is no need to compromise the story for the group. Once you get the hang of duets they really are a wonderful experience, and for some players infinitely preferable to groups.