The accurate prediction of ligand-biopolymer binding affinities is of general interest to medicinal chemistry, as well as to the broader field of molecular recognition. The ability to predict computationally the thermodynamics of these molecular recognition processes has been relatively weak until recently, however, continued developments on several fronts are extending the scope of applicability of these methods. The rapid growth in the number of protein-ligand structures has initially led to the development of a range of empirical scoring functions based on relatively simple descriptions of intermolecular interactions. These methods have had some success in ranking binding affinities when tuned to particular protein systems or in rather qualitative estimates of molecular fit in fast docking calculations. However, they are too unreliable for more detailed, quantitative, assessment and comparison of binding affinities. Physics-based free energy calculations are in principle more general and have the potential to be significantly more accurate. These approaches have seen steady development over many years and rely on carefully calibrated molecular energy functions (force-fields), simulations of the systems with explicit solvent, and the coming-of-age of continuum solvation models. In addition to the initially developped Free Energy Perturbation (FEP) and Thermodynamic Integration (TI) methods, new approaches include the Molecular Mechanics-Poisson-Boltzmann Surface Area (MM-PBSA) and the Linear Interaction Energy (LIE) approaches. This review concentrates on MM-PBSA and LIE, and their variants. The routine application of these calculations is becoming possible because of enhanced computational hardware and the development of a range of computational chemistry tools. This review addresses: i) the basic principles behind free energy calculations ii) recent methodological advances iii) comparisons of predicted and experimentally determined affinities iv) the uncertainties and limitations of both the computational and experimental data v) areas where progress can be made vi) the practicality of applying the methods at the different stages of the drug discovery and optimization process.