This thesis concerns the problem of classifying blocks with an elementary abelian defect group, and in particular blocks with an elementary abelian defect group of order 32. In Chapter 1 we establish the notation and introduce the key objects and arguments on which modular representation theory and block theory are built. In Chapter 2 we introduce (G, B)-local systems and crossed products, which we use to investigate block covering relations, and we describe a general method that can be used to classify blocks once a specific list of prerequisites has been achieved. In Chapter 3 we employ this method to classify blocks with an elementary abelian defect group of order 32. Due to the lack of all necessary prerequisites, we employ alternative, more ad-hoc techniques to obtain the result. These techniques, while less general, still have good potential to be useful in many more other cases. In Chapter 4 we examine the case of blocks with an elementary abelian defect group of order 64, and we classify all principal blocks with such defect group.