The initial-boundary value problem for a system of nonlinear parabolic equations modeling the growth and migration of glioma cells is studied. New a priori estimates of the solution are obtained, based on which the non-local in time unique solvability of the initial-boundary value problem is proved. The conducted computational experiments demonstrate the ability of the mathematical model to reflect the desired properties like development of hypoxia in the tumor microenvironment, the switching of proliferative tumor cells to invasive migratory ones due to hypoxia, the regression of the vasculature in the area occupied by the tumor and simultaneous angiogenesis at the periphery of the tumor.