This paper considers three challenge areas for mechanism design and describes the role approximation plays in resolving them. Challenge 1: optimal mechanisms are finely tuned to precise details of the distribution on agent preferences. Challenge 2: in environments with multi-dimensional agent preferences economic analysis has failed to provide general characterizations optimal mechanisms. Challenge 3: optimal mechanisms are parameterized by unrealistic knowledge of the distribution of agents' private preferences. This paper surveys positive resolutions to these challenges with emphasis on basic techniques and their relevance to theory and practice.