Individual-based models have been applied to study a broad spectrum of problems across multiple disciplines, such as the spread of epidemics or the outcome of social dilemma. They are used to investigate the macroscopic effects that arise from the microscopic dynamics of interacting individuals. Fixation describes the taking over of the population by a single type of individual or species. It is a prominent feature in the field of population genetics, which interprets many biological scenarios of evolution. Equilibration describes the process of reaching a heterogeneous steady state. In this thesis we analyse these macroscopic features through techniques derived from statistical physics and the theory of stochastic processes.Birth--death processes are used to describe the interaction of two types of individual in a population, such as competing strains of bacteria. These interactions are often specified using the framework of evolutionary game theory. The environment in which the population evolves can have a crucial impact on selection. In systems where the environment switches between multiple states we develop a general theory to calculate the fixation time statistics of a mutant individual in a population of wild-types, as well as the stationary distributions when mutations are present in the dynamics. In some birth--death processes, and in particular those described by evolutionary game theory, the mean fixation time contains only limited information. By diagonalising the master equation that describes the process, we are able to obtain closed-form expressions for the complete fixation time distributions. Individual-based models can also be used to describe the accumulation of mutations in a cell. This has important consequences for the initiation and progression of cancer. We find that such systems exhibit metastable states in the dynamics, and we can exploit the separation of timescales between relaxing to the quasi-stationary state and reaching fixation to characterise these phenomena. In this scenario we employ the WKB method to describe the population-level dynamics. Although this method has been used to describe numerous stochastic processes, a clear and coherent description is lacking in the literature. Through the use of multiple examples, including the aforementioned cancer initiation model, we carefully explain the multitude of constructs and equations that result from the application of this methodThe analytical characterisation of the evolutionary dynamics that are observed in these stochastic processes has resulted in a greater understanding of fixation and equilibration. This thesis promotes the benefits of analytical, or even semi-analytical methods, and on a more general level contributes toward a more complete understanding of evolutionary processes.