Graphs are a ubiquitous data structure in computer science and can be used to represent solutions to difficult problems in many distinct domains. This motivates the use of Evolutionary Algorithms to search over graphs and efficiently find approximate solutions. However, existing techniques often represent and manipulate graphs in an ad-hoc manner. In contrast, rule-based graph programming offers a formal mechanism for describing relations over graphs. This thesis proposes the use of rule-based graph programming for representing and implementing genetic operators over graphs. We present the Evolutionary Algorithm Evolving Graphs by Graph Programming and a number of its extensions which are capable of learning stateful and stateless digital circuits, symbolic expressions and Artificial Neural Networks. We demonstrate that rule-based graph programming may be used to implement new and effective constraint-respecting mutation operators and show that these operators may strictly generalise others found in the literature. Through our proposal of Semantic Neutral Drift, we accelerate the search process by building plateaus into the fitness landscape using domain knowledge of equivalence. We also present Horizontal Gene Transfer, a mechanism whereby graphs may be passively recombined without disrupting their fitness. Through rigorous evaluation and analysis of over 20,000 independent executions of Evolutionary Algorithms, we establish numerous benefits of our approach. We find that on many problems, Evolving Graphs by Graph Programming and its variants may significantly outperform other approaches from the literature. Additionally, our empirical results provide further evidence that neutral drift aids the efficiency of evolutionary search.