The present paper is intended to give a survey of the developments of the wavelet Galerkin boundary element method. Using appropriate wavelet bases for the discretization of boundary integral operators yields numerically sparse system matrices. These system matrices can be compressed to O(N_j) nonzero matrix entries without loss of accuracy of the underlying Galerkin scheme. Herein, O(N_j) denotes the number of unknowns. As we show in the present paper, the assembly of the compressed system matrix can be performed within optimal complexity. By numerical experiments we provide examples which corroborate the theory.