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Within the framework of equilibrium statistical mechanics the free energy of a phase gives a measure of its associated probabilistic weight. In order to determine phase boundaries one must then determine the conditions under which the free energy difference (FED) between two phases is zero. The underlying complexity usually rules out any analytical approaches to the problem, and one must therefore adopt a computational approach. The focus of this thesis is on (Monte Carlo) methodologies for FEDs. In order to determine FEDs via Monte Carlo, the simulation must (in principle) be able to visit the regions of configuration space associated with both phases in a single simulation. Generally however one finds that these regions are significantly dissimilar, and are separated by an intermediate region of configuration space of intrinsically low probability, so that a simulation initiated in either of the phases will tend to remain in that phase. This effect is generally referred to as the overlap problem and is the most significant obstacle that one faces in the task of estimating FEDs. In chapter 2 we start by formulating the FED problem using the Phase Mapping (PM) technique of [1]. This technique allows one to circumvent the intermediate regions of configuration space altogether by mapping configurations of one phase directly onto those of the other. Despite the improvement that one gets when formulating the problem via the PM, the overlap problem persists, albeit to a lesser degree. In chapter 2 we define precisely what we mean by overlap and then discuss a range of methods that are available to us for calculating the FEDs within the PM formalism. In the subsequent chapters we then focus on the three generic strategies that arise in addressing the overlap problem. The first strategy that we focus on (chapter 3) is that of the representation. This corresponds to the choice of coordinate system with which one parameterises the degrees of freedom of the two phases. The PM works by matching these coordinates in the two systems, and therefore its efficiency in generating overlap is dependent on the choice of representation. We examine a particular representation in which the PM matches the fourier coordinates of the two phases in such a way so as to achieve perfect overlap in the harmonic limit (for structurally ordered phases). Previous formulations [1] have been limited to the realspace PM (RSM). By comparing the RSM to the fourier version (FSM), we show that for a range of temperatures the overlap associated with the FSM is considerably better than that of the RSM, thus allowing one to determine the FED efficiently under conditions in which the RSM would otherwise fail. The second strategy that we study (in chapter 4) is that of the estimator which one uses to determine the free energy difference. The different estimators pool the data from the various regions of configurations space in different ways, and therefore have different (systematic and statistical) errors associated with them. As a consequence the severity of the overlap problem is dependent on the estimator that one employs. We examine conditions under which the different estimators are able to arrive at estimates which are free of systematic errors. 1