The emergence of drug resistance is ultimately driven by Darwinian evolution. These evolutionary dynamics are inherently two-tiered, with mutational processes at the genetic scale inducing variation in cellular phenotypes that are subject to natural selection. If we are to predict or reverse evolution, as we must to determine effective treatments for drug-resistant infections and cancers, then we must first understand the relationship between genetic and phenotypic change. This relationship, known as the genotype-phenotype (GP) mapping, is governed by a complex cascade of potentially stochastic molecular interactions that integrate genetic, epigenetic and environmental factors to determine cellular phenotype. In this thesis, we introduce abstract models of the GP-mapping to explore how its structure influences the evolution of drug resistance, and determines the efficacy of novel therapeutic strategies such as drug holidays or adaptive therapy. We begin by providing a comprehensive review of previous abstract modelling of the GP-mapping (Chapter 2). In Chapter 3, we demonstrate, by means of a fitness landscape model for the GP-mapping, that through careful selection of drug sequences evolution can be `steered' such that a highly drug resistant population does not emerge. In Chapter 4, we introduce a novel model for the GP-mapping wherein phenotypes are determined by the stochastic simulation of bistable chemical reaction networks (CRNs). Through this model we explore the evolutionary dynamics of ecological `bet-hedging', a common driver of drug resistance wherein phenotypes are determined stochastically, demonstrating that the structure of the GP-mapping can slow the evolutionary loss of this trait, preserving the survival mechanism when harsh environments occur very infrequently. Thus, the capacity to steer the evolutionary loss of bet-hedging through drug holidays is dependent on the structure of the GP-map. In Chapter 5, the CRN model is extended to account for epigenetic inheritance and the potential for phenotypic memory in bet-hedging. Critically, we find that genetics, epigenetics and the GP-mapping interact non-additively to determine organismal fitness, indicating that evolution likely cannot be predicted without accounting for each of these biological processes. Finally, in Chapter 6, we explore how properties of the GP-mapping are manifested at the population scale and suggest experimental approaches to identify bet-hedging and phenotypic plasticity from population measurements. The implications of our results for the treatment of drug-resistant diseases are explored throughout.