The Klein-Gordon equation without dispersion, and with quadratic and cubic nonlinearities, has been studied in one and higher dimensions. Algebraic solitary wave solutions in all cases, as well as higher-order modes in higher dimensions (similar to nonlinear optics) have been shown to exist corresponding to specific initial values. While in the one-dimensional case, arbitrary initial values yield periodic solutions, asymptotically stable solutions are shown to exist in the higher-dimensional case. For both one- and higher-dimensional cases, solutions tending to zero with distance are shown to be achieved for other initial conditions by incorporating a small amount of 'saturating' fourth-order nonlinearity. Finally, it is shown how a general Klein-Gordon equation with dispersion and a forcing term may be reduced to the equation discussed in the paper.