In the past few years we have seen that the bootstrap approach to higher-dimensional conformal field theories (CFTs) can be surprisingly powerful. In particular, we are finally able to put the crossing symmetry equations to good use and extract nontrivial information about the spectrum and three-point functions in a generic CFT. In this talk I will discuss the application of these ideas to superconformal field theories, focussing on N=2 and N=4 theories in four dimensions. In those theories there exists a subsector where the crossing symmetry equations can be solved analytically. Together with the numerical analysis of the remaining constraints we can learn a great deal about the nonperturbative structure of these superconformal field theories.