The energy of any atom or molecule with an even number N of electrons is shown to be an exact functional of a single geminal where the functionals for both the kinetic energy and the external potential are explicitly known. We derive the foundations for geminal functional theory (GFT) through a generalized constrained search and the use of two theorems which demonstrate that all one-particle properties of atoms and molecules with even N may be parametrized by a single geminal [A. J. Coleman, Int. J. Quantum Chem. 63, 23 (1997); D. W. Smith, Phys. Rev. 147, 896 (1966)]. By generalizing constrained search to optimize the universal functionals with respect to the 2-RDM (two particle reduced density matrix) rather than the wave function, we closely connect the one-density, the 1-RDM (one-particle reduced density matrix), and the geminal functional theories with 2-RDM minimization of the energy. Constrained search with the 2-RDM emphasizes that all energy functional methods must implicitly account for the N-representability of the 2-RDM within their universal functionals. An approximate universal functional for GFT, equivalent to a variational ansatz using the antisymmetrized geminal power wave function, yields energies that are significantly better than those from Hartree-Fock and yet rigorously above the exact energy.