The thermodynamic properties and phase transformations of metals and alloys are studied using the statistical moment method, going beyond the quasi-harmonic approximations. Including the power moments of the atomic displacements up to the fourth order, the Helmholtz free energies and the related thermodynamic quantities are derived explicitly in closed analytic forms. The thermodynamic quantities, like thermal lattice expansion coefficients, specific heats, Grüneisen constants, elastic constants calculated by using the SMM are compared with those of other theoretical schemes and the experimental results. The hcp-bcc structural phase transformations observed for IVB elements, Ti, Zr and Hf, are discussed in terms of the anharmonicity of thermal lattice vibrations. The equilibrium phase diagrams are calculated for the refractory Ta-W and Mo-Ta bcc alloys. In addition, the temperature dependence of the elastic moduli C11, C12 and C14 and those of the ideal tensile and shear strengths of the bcc elements Mo, Ta and W are studied: We also discuss the melting transitions of metals and alloys within the framework of the SMM and estimate the melting temperatures through the limiting temperature of the crystalline stability.