Adaptive discretizations form a key methodology for treating large scale numerical simulation problems derived from partial differential or singular integral equations. This talk highlights some recent developments centering upon adaptive wavelet schemes. It is shown how concepts from nonlinear or best N-term approximation and computational Harmonic Analysis can be used to design and analyse adaptive schemes with asymptotically optimal complexity for a wide class of stationary linear and nonlinear operator equations. Special emphasis is put on the adaptive approximate application of linear and nonlinear operators in wavelet coordinates. It is indicated how numerical schemes can be derived from quantitative estimates for scale separation.