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If G is a finite group and p a prime dividing the order of G, Sylow's Theorem tells us that G has a p-subgroup of largest possible order, and that such a group is essentially unique. This begins the study of the p-local structure of finite groups: That part of that group visible to a particular prime. These data are organized in a category called a ''fusion system''. The p-local study of finite groups has been an area of great interest to group theorists for decades, but more recently connections to algebraic topology have surfaced as well. The Martino-Priddy Conjecture states that two groups have p-equivalent classifying spaces if and only if their fusion systems are isomorphic. In the course of proving this result, it became necessary to think of fusion systems as algebraic objects in their own right, separated from the finite groups in which they originally arose. Following Puig and Broto-Levi-Oliver, once one has identified the key p-local structures of a finite group, one has introduced a new type of algebraic object, a ''p-local finite group''. These are group-like algebraic objects that only exist at a particular prime. p-local finite groups can be seen as more general than finite groups, which live in all primes at once. In this talk we will introduce these concepts and give a brief overview of how fusion theory serves as a bridge between finite groups, algebraic topology, and representation theory.