The regularity problem in CR geometry

Given two smoothly bounded domains in complex Euclidean spaces, possibly of different dimensions, we can ask whether any holomorphic map between them, which maps the domain boundary of the first into that of the second, must extend to a smooth map of the boundaries. The quest for local methods to deal with this problem leads to the regularity problem in CR geometry. We will first skim through a necessary amount of CR geometry and then discuss some recent results on the regularity problem.