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The free energy perturbation method, and the related thermodynamic integration methods, have recently had a great influence on theoretical chemists interested in applying computational techniques to study problems of biological importance. The infinite order free energy perturbation equation was derived almost forty years ago, but only within the last five years has it been widely used in conjunction with Monte Carlo or molecular dynamics simulations to calculate free energy differences between similar macromolecular systems in solution. The purpose of this review is firstly to describe the formalism behind the methods but secondly and more importantly to describe the wide range of problems to which it has been applied. Here the focus is primarily on problems of chemical or biological interest rather than on problems studied for the sake of methodological development. The methods, of necessity, are generally …