The Finite Point Method (FPM) is a meshless technique which is based on both, a Weighted Least-Squares numerical approximation on local clouds of points and a collocation technique which allows obtaining the discrete system of equations. The research work we present is part of a major investigation into the capabilities of the FPM to deal with threedimensional applications concerning real compressible fluid flow problems. In the first part of this work, the upwind biased scheme employed for solving the flow equations is described. Secondly, with the aim of exploiting meshless capabilities, an h-adaptive methodology for two and three-dimensional compressible flow calculations is developed. This adaptive technique applies a solution-based indicator in order to identify local clouds where new points should be inserted in or existing points could be safely removed from the computational domain. The flow solver and the adaptive procedure have been evaluated and the results are highly encouraging. Several numerical examples are provided throughout the article in order to illustrate their performance.