The formulation and implementation of wavelet based methods for the solution of multi-dimensional partial differential equations in complex geometries is discussed. Utilizing the close connection between Daubechies wavelets and finite difference methods on arbitrary grids, we formulate a wavelet based collocation method, well suited for dealing with general boundary conditions and nonlinearities. To circumvent problems associated with completely arbitary grids and complex geometries we propose to use a multi-domain formulation in which to solve the partial differential equation, with the ability to adapt the grid as well as the order of the scheme within each subdomain. Besides supplying the required geometric flexibility, the multidomain formulation also provides a very natural load-balanced data-decomposition, suitable for parallel environments. The performance of the overall scheme is illustrated by solving two dimensional hyperbolic problems. (C) 1998 Academic Press.