Classical investigations on the ellipsoidal figures of equilibrium of liquid masses are here enlarged by Chandrasekhar into a complete theory. The author develops and completes the basic ideas put forth in three fundamental papers by Dirichlet, Dedekind, and Riemann over a century ago, which have been all but ignored since that time. The various problems are solved by a method and a technique that are essentially elementary, and a number of common misconceptions and errors are corrected. After a historical introduction, the author goes on to discuss virial equations of the various orders and to describe his new method; potentials of homogeneous and heterogeneous ellipsoids (including theorems on a class of heterogeneous ellipsoids which enable the treatment of the subject without explicit use of ellipsoidal harmonics); Dirichlet's problem and Dedekind's theorem; Maclaurin spheroids; Jacobi and Dedekind ellipsoid; Riemann ellipsoids; Roche ellipsoids (Including Darwin ellipsoids).