Coordinate system selection is an important consideration in the asymptotic numerical solution of any fluid flow or heat transfer problem. Ideally, the mesh point distribution should be such that the error in the computed solution is minimized. Since the error depends on the solution itself, the point distribution should constantly change to suit the changing solution. A grid that automatically adjusts itself to the solution being calculated is called an adaptive grid. Three different adaptive grid procedures have been developed in this study. Two of these schemes help in properly distributing points to reduce the error in the computed solution. In shock capturing applications, a grid that is aligned with the shock is a must for obtaining dispersion error free solutions. The third adaptive grid scheme helps to align coordinate lines with discontinuities in the flow field;Many researchers have worked on the point redistribution idea in the past. The methods that they have developed are applicable only to simple one-dimensional or quasi one-dimensional problems. The methods developed in this study have been applied to fully two-dimensional calculations involving systems of partial differential equations, curved boundaries and stationary and nonstationary boundaries. The extension of the adaptive grid procedures developed here to three-dimensional problems is trivial;The point clustering schemes have been applied to Burgers' equation in one and two dimensions, the inviscid supersonic flow over wedges and cylinders with the associated detached bow shocks, the laminar boundary layer over a flat plate and to simple one-dimensional calculations involving flow variable discontinuities. The aligning scheme has been applied to the problem of a straight oblique shock in a rectangular region and the shocked flow through an expanding two-dimensional duct. Substantial reductions in error are demonstrated in all cases with the use of the adaptive grids.