Some of the results of a rigorous analysis of the Chandrasekhar (1931 and 1984) semiclassical theory of stellar collapse are presented here. They are of two kinds. The first concerns the Chandrasekhar equation itself; the uniqueness of the solution is proven, and also the fact that the solution has certain properties not noted before. The second is a derivation of the Chandrasekhar equation from quantum mechanics (in the limit of small gravitational constant) without making a priori assumptions about the smallness of correlation effects. A parallel derivation is made for stars composed of bosons (such as axion stars, if they are ever found to exist); the resulting equation is quite different from the Chandrasekhar equation, for it is of the Hartree type and involves density gradients.