We give a survey on recent developments on wavelet-based numerical solution of time-dependent partial differential equations. The fundamental idea is to use wavelets to give sparse representations of the solution operators involved. Thus it leads to a fast algorithm for efficient approximation of the solution to the PDE. Such an operator representation also applies to a large class of integral and differential operators, including the Calder\'on-Zygmund operators and pseudo-differential operators. We demonstrate the general scheme by considering the anisotropic diffusion equation modeling arising in thin film image processing. Among other examples are advection-diffusion equations arising in CFD. Numerical results are presented.