Despite our well-advanced understanding of finite groups, for the group isomorphism problem major challenges remain: 1.) the complexity of the group isomorphism problem, which forms the foundation to designing efficient algorithms, is unknown, 2.) current implementations for group isomorphism are based on heuristics and no thorough analysis or comparison of algorithmic approaches is available, and 3.) we lack an understanding which properties of a group govern its structure as a whole. The project addresses these challenges by studying the computational complexity, developing and testing new group isomorphism algorithms, and pinpointing combinatorial properties that determine the structure of a group.