The use of grid adaptation procedures for compressible flows and heat transfer is presented in this paper. Grid adaptation schemes are presented for structured and unstructured grids in the context of finite-difference and finite-element methods. For compressible flow problems with regular domains, the use of finite-difference methods has proven to be quite effective. For such analyses, a dynamic grid adaptation scheme is employed with a MacCormack predictor-corrector scheme to describe compressible flow features. The adaptation procedure uses a grid relocation stencil that is valid at both the interior and boundary points of the finite-difference grid. Linear combinations of spatial derivatives of specific flow variables, calculated with finite-element interpolation functions, are used as adaptation measures. The impact of the grid relocation procedure is demonstrated by simulating the effects of shock impingement on a three-dimensional unsymmetric axial corner. When the domain to be modeled is quite complex, unstructured finite-elementor finite-volume formulations are preferred. An unstructured mesh adaptation scheme with refinement and coarsening based on refinement indicators is used to describe inviscid and viscous flow features. This adaptation scheme uses quadrilateral and triangular elements to add grid density to regions of high gardients and remove elements where these gardients are low. Modifications of the mesh adaptation procedure enables its utilization in the analyses of heat transfer problems commonly associated with hypervelocity flows. The strategies used for mesh adaptation, computing refinement indicators, and time marching are described. The effectiveness of this procedure is reflected in good solution accuracy, reduction in the number of elements used, and computational efficiency.