A dynamic solution adaptive mesh method was implemented into a finite-volume numerical method for solving unsteady flowfields described by the two-dimensional, unsteady, Navier-Stokes and Euler equations. The objective was to improve the resolution and accuracy of solutions which contained flow gradients which varied in strength and position with time. Variational principles were used to formulate the mesh equations with which meshes were generated to have the desired smoothness, orthogonality, and volume adaption qualities. The adaption of the mesh to the flow solution was driven by the presence of flow gradients. The dynamics of the mesh was accounted for in the flow equations through the mesh speeds. A comparison was made between one approach which computed the mesh speeds from a backwards time differences of the mesh and another approach which computed the mesh speeds from a system of mesh speed equations which were derived from the time differentiation of the mesh equations. The dynamically adaptive mesh method was demonstrated for model problems involving solution and boundary dynamics, inviscid flows in a converging-diverging nozzle, viscous boundary-layer flows over flat plates, and viscous flows in a transonic diffuser. It was found that the approach using the mesh speed equations was more accurate than the approach using the time-differenced mesh speeds. There was difficulty is obtaining proper clustering of the meshes for viscous flows.