An adaptive finite-volume numerical method has been developed for the unsteady Navier-Stokes equations of incompressible flow in two dimensions. The momentum equations combined with a pressure correction equation are solved employing a non-staggered grid. The solution is advanced in time with an explicit marching scheme. An adaptive algorithm has been developed, which refines the grid locally in order to resolve detected flow features. A combination of quadrilateral, as well as triangular cells provides flexibility in forming the adaptive grids. Treatment of the grid interfaces proved flexibility in forming the adaptive grids. Treatment of the grid interfaces proved to be stable. Applications of the developed adaptive algorithm include both steady and unsteady flows with different Reynolds numbers. Comparisons with analytical, as well as experimental data evaluate accuracy and robustness of the method.