Over the past few decades, density functional theory (DFT) [1-4] methods have demonstrated high accuracy in calculations of structural and energetic properties in a wide variety of materials. While calculations at absolute zero have traditionally been the focus of this ab initio (or ”rst-principles) literature, there is considerable and growing interest in obtaining ”nite-temperature properties, such as equilibrium phase diagrams and, more generally, thermodynamic properties. The main dif”culty associated with this endeavor is that highly accurate quantum-mechanical electronic structure calculations are currently limited in terms of size of the simulation cell and in the number of distinct atomic con”gurations that can be feasibly sampled within a reasonable computing time frame. On the other hand, calculations of thermodynamic properties require, by de”nition, large system sizes and extensive sampling of phase space. Fortunately, this impasse can be resolved by coupling DFT energy methods with statistical mechanical models [5-14]. This chapter provides an overview of the methodologies underlying ”rst-principles calculations of alloy phase stability and provides examples intended to illustrate the capabilities and accuracy of these methods.