Abstract <p>Multidimensional nonlinear Schrödinger equations of general form with potential and dispersion specified by one or two arbitrary functions are studied. The equations under consideration naturally generalize a number of related nonlinear partial differential equations encountered in different areas of theoretical physics, including nonlinear optics, superconductivity, and plasma physics. One- and multidimensional non-symmetry reductions leading the studied nonlinear Schrödinger equations to simpler equations of lower dimensions or ordinary differential equations (or systems of ordinary differential equations) are described. Particular attention is paid to the search for solutions with radial symmetry. Using methods of generalized and functional separation of variables, new exact solutions are found in quadratures or elementary functions for two- and <i>n</i>-dimensional Schrödinger equations of general form.</p>