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Biological evolution is an inherently non-equilibrium process, by which a population acquires a new genetic composition, optimally suited to its present environment. Far from being the slow process it is traditionally viewed as, the rapid evolution of microbes is causing serious global concern in the acquisition of microbial resistance to antibiotics. Better understanding of the mechanisms that govern the evolution of microbes is therefore of paramount importance. In many traditional models, evolution occurs over the space of all possible genetic states (genotypes). These are assigned a quantity called fitness, which quantifies that genotype's suitability over others to thrive within its present environment. A population of replicating cells can evolve over this space under the competing influences of random variations of the genotype (i.e. mutations) and the increased likelihood of success for fitter genotypes (i.e. selection). Many of these models fail to account for the observation that biological diversity is rife, even amongst genetically identical cells that exist in the same environment. This diversity manifests itself as a difference in phenotype (the observable traits of an organism). It means that organisms with the same genotype, but a different phenotype, may have different fitnesses. Therefore, when phenotypic heterogeneity is apparent, evolution over genotype space should consider different fitness landscapes for each of the distinct phenotypic states that exist. Phenotypic heterogeneity has long been observed in populations of microbes. Often these can switch between different phenotypic states for a number of reasons. A common example of this is stochastic phenotype switching, in which cells randomly switch between two phenotypic states, without any inducing influence. This has been shown to benefit populations of cells that are subject to fluctuating environmental conditions, or by creating a division of labour in the population. In this work, I examine the possibility of another role for stochastic phenotype switching: as a mechanism that can accelerate evolution even in a static environment. During evolution, populations can spend large amounts of time trapped at local peaks on a fltness landscape. A cell that switches phenotype will change to a different fitness landscape, which may allow for faster genetic evolution. I begin this work in Chapters 3 and 4, where I present a model of an evolving population of haploid cells, trapped at a local peak on a 1D fitness landscape. These cells have access to a second phenotypic state, in which the fitness landscape is uniform. The focus of this study is to see the effect that stochastic phenotype switching to this secondary phenotype has on the populations evolution of a target state. In Chapter 3 I study this numerically and identify an optimal range for the rate of phenotype switching, within which the time taken for the process can be reduced by many orders of magnitude. I also find that if the frequency of switching is allowed to evolve, then the likely evolutionary trajectory taken by a population is one that first evolves a switching frequency to within the identified optimal range, before escaping from the local peak. In Chapter 4 I present an analytic study of the same model. The aim here is to recover the numerical results from Chapter 3. I employ numerous analytic techniques to show the existence of the optimal range, while developing an analytic approach that allows a study of the model at parameter values that are otherwise difficult to simulate. This same model is extended in Chapter 5 to consider evolution over a more complex genotype space: that of a hypercube. Here, genotypes correspond to particular binary sequences, which can be used as representations of many biological states of interest; for example, nucleotide sequences in DNA or the presence and absence of important mutations in specific genes. My focus here is again on the effect that stochastic phenotype switching has on how a population of cells evolves over genotype space. This is studied numerically for various kinds of randomly generated fitness landscapes. I find that in some instances phenotype switching can significantly benefit a population. However, in other instances it can significantly hinder the evolution, increasing the time taken for the process by many orders of magnitude. Finally, in Chapter 6 I present a model that explores how a population of the bacterium Escherichia coli (E. coli) evolves resistance to the antibiotic ciprofloxacin. This work is motivated by the observed rapid acquisition of resistance of E. coli when exposed to sub-lethal concentrations of the antibiotic. Upon damage to their DNA, cells can induce a switch to a secondary phenotypic state (as part of the SOS response), in which DNA repair and an increased rate of mutations occur. Using this model, with empirical data for the fitness and susceptibility of genotypes, I numerically explore the dependence of rapid evolution on the existence of this secondary phenotypic state. I find that the model predicts, over the short timescales considered, that the evolution of sufficient resistance requires the existence of the secondary phenotypic state. The findings of this work is that the phenotypic switching of cells can have a significant impact on how populations evolve in static environments. While stochastic phenotype switching can help populations escape from local peaks, it can also trap populations on sub-optimal landscapes if the frequency of switching is too low.