We review the Anderson and Pulay acceleration methods which are also known as the direct inversion of iterative subspace (DIIS) methods for solving nonlinear equations and show how they are used in electronic structure calculations to solve the Kohn-Sham equation and the coupled cluster equation. We present numerical examples to compare the performance of acceleration methods with conventional Quasi-Newton and Newton Krylov methods. We also show that these acceleration methods can be used effectively to refine eigenvector approximations in iterative methods for solving linear eigenvalue problems.