We consider the problem of triangulating a d-dimensional region. Our mesh generation algorithm, called QMG, is a qradtree-based algorithm that can triangulate any polyhedral region including nonconvexregions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles. Finally, our algorithm is guaranteed never to over refine the domain in the sense that the number of simplices produced by QMG is bounded above by a factor times the number produced by any competing algorithm, where the factor depends on the aspect ratio bound satisfied by the competing algorithm. The QMG algorithm has been implemented in C++ and is used as a mesh generator for the finite element method.