In this dissertation we deal mainly with the problem of metastability in statistical mechanics. In Chapter I we review the analytic aspects of metastability. Taking stability as the basic concept Sewell defined metastable states to be those states which are locally but not globally stable. For a hard core continuous system the pressure functions of the metastable state is found to be the real analytic continuation in the chemical potential of the pressure for the equilibrium phase In Section 2 we present the result of Landford and Ruelle showing the impossibility, in a model with short range forces, of the existence of metastable states. Langer Vs method using the Mayer series expansion ends this section. We also discuss two descriptions of metastability based on eigenvalue degeneracy. In Chapter II we describe the dynamic aspects of metastability. A "restricted ensemble" is defined and it is proved that the escape rate can be made very small. By using the Becker- Dôring set of equations it is shown that the behaviour of the system is governed by a "slow mechanism" if the state is a point in the restricted ensemble. We provide a rigorous version of the Boltzmann-factor approximation to the expected number of N-particle clusters mN. Similar results are shown to hold in N a special case of an Ising model and in a hard core continuous model. In Chapter III we work with a continuous model and obtain an approximation to mN which is equivalent to the one obtained in Chapter II; by making reasonable assumptions we present an expression for mN in agreement with the results of nucleation theory.