Non-Gaussian random-matrix ensembles are important in many applications. We propose Monte Carlo and Langevin methods for generating non-Gaussian ensembles and their eigenvalue spectra. We also provide a general framework for analytic studies of the level density in these ensembles. We show that, in general, the level densities exhibit banded spectra, with important implications for mesoscopic systems and complex nuclei. The universality of energy-level fluctuations is confirmed.