Mathematically rigorous versions of Thomas-Fermi Theory and its generalizations were developed in the 1970's and 1980's by Lieb, Simon, Benilan, Brezis, Gallouet, Morel and others. At issue is the electron density for an N electron quantum mechanical system in its ground state. The energy minimization problem is reduced to the solution of the Euler-Lagrange equation, which is reduced to the solution of a nonlinear elliptic equation in L('1) The theory which will be presented includes extensions of the existing theory to d ((GREATERTHEQ)3) dimensions and the introduction of weight functions into the kinetic energy term. Existence, nonexistence, and uniqueness results will be presented, as well as qualitative properties of the solutions