We discuss the generation of curvilinear composite overlapping grids and the numerical solution of partial differential equations on them. A composite overlapping grid consists of a set of curvilinear component grids that cover a region and overlap where they meet. Continuity conditions (interpolation) are imposed at the overlapping boundaries. The principal advantage of composite grids is in the generation of grids for regions with complicated geometries. The grid construction program CMPGRD is used to create composite grids with any number of component grids. We describe some techniques for the solution of elliptic and time-dependent PDEs on composite meshes. Applications to the solution of the compressible Navier-Stokes equations are presented. © 1990.