Wavelet bases are offering an efficient way for the numerical approximation of functions and operators, differential- as well as integral operators. From theoretical and practical perspectives the potential of these method has to be explored. We remark here important norm characterizations and their impact for an adaptive and efficient approximation of functions and (non local) operators. Beside theoretical results, we would like to present several numerical examples in which wavelet bases are applied for an efficient solution of problems in numerical analysis and engineering. These examples taken are from boundary integral equations, inverse problems and some particular problems concerning the numerical solution of partial differential equations.