A review is given of the current mathematical and theoretical status of the possibility of the formation of binary stars by fission. It is emphasized that none of the competing theories is completely deductive: each begins with a preconception of how fission might occur, and considers mathematical problems that bear on the viability of that preconception. The classical fission theory turned out to lack internal consistency. The present author’s version of the theory, revised from the classical theory in that viscosity plays a negligible rather than a dominant role, appears to be internally consistent and compatible with the idea that close binaries with mass ratios near one may form in this way. The version of the theory described elsewhere in this volume by Durisen, which fails to produce binaries, is based on a preconception of the route to fission different from the classical one. Its failure to produce binaries therefore reflects only on that route, and not on other possible routes to fission.