The first type of result shows nonapproximability: for many problems, computing approximate solutions is no easier than computing optimal solutions. This is shown by proving a new probabilistic characterization of NP, according to which it is possible to check certificates for membership in an NP language by examining a constant number of bits in the certificate. We will survey this result (the so-called PCP Theorem) and its many extensions.
The other type of result consists of designing better approximation algorithms. This endeavor has also been successful for many problems, and a large body of techniques has been developed in the past decade. We will survey some of these.