The Hartree-Fock exchange operator is an integral operator arising in the Hartree-Fock method and replaced by a multiplicative operator (a local potential) in Kohn-Sham density functional theory. This article presents a detailed analysis of the mathematical properties of various local approximations to the nonlocal Hartree-Fock exchange operator, including the Slater potential, the optimized effective potential (OEP), the Krieger-Li-Iafrate (KLI) and common energy-denominator approximations (CEDA) to the OEP, and the effective local potential (ELP). In particular, we show that the Slater, KLI, CEDA potentials and the ELP can all be defined as solutions to certain variational problems. We also provide a rigorous derivation of the integral OEP equation and establish the existence of a solution to a system of coupled nonlinear partial differential equations defining the Slater approximation to the Hartree-Fock equations.