In this chapter, one aim is to study spaces of mappings taking their values in a Lie group. It will turn out that these spaces carry again a natural Lie group structure. However, before we prove this, the definition and basic properties of (infinite dimensional) Lie groups and their associated Lie algebras are recalled. Infinite-dimensional Lie theory (beyond Banach spaces) is by comparison relatively young and in its modern form goes back to Milnor’s seminal works. One key feature of infinite-dimensional Lie theory is that the conncection between Lie algebra and Lie group is looser then in finite dimensions. For advanced tools in Lie theory one has to require the Lie group to be regular (in the sense of Milnor). These concepts are introduced and considered for several main classes of examples, such as the diffeomorphism groups, loop groups and gauge groups.