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Loop groups (spaces of maps $S^1 \to G$ where $G$ is a Lie group) are nice examples of infinite dimensional manifolds, and they are important in physics and string theory. They have a rich and interesting theory, the basics of which are developed for example in the book "Loop Groups" by Pressley and Segal. See also the paper "Unitary representations of some infinite dimensional groups" by Segal. Moreover Freed-Hopkins-Teleman have many interesting results which relate the the representation theory of the loop group $LG$ and the twisted equivariant $K$-theory of the Lie group $G$.