In this thesis, we apply model theory to Lie theory and geometric group theory. These applications of model theory come via nonstandard analysis. In Lie theory, we use nonstandard methods to prove two results. First, we give a positive solution to the local form of Hilbert's Fifth Problem, which asks whether every locally euclidean local topological group is locally isomorphic to a Lie group. In connection with the local form of Hilbert's Fifth Problem, we study local groups with a local automorphism whose iterates pointwise approach the trivial endomorphism. Secondly, we prove a generalization of a theorem of Pestov regarding Banach-Lie algebras. Call a Banach-Lie algebra "enlargeable" if it is the Lie algebra of a Banach-Lie group. Pestov used nonstandard methods to prove that a Banach-Lie algebra is enlargeable if it possesses a directed family of "uniformly enlargeable" subalgebras whose union is dense. We prove an analogue of this result for a wider class of infinite-dimensional Lie algebras, namely the "locally exponential Lie algebras." In geometric group theory, we give a nonstandard treatment of the theory of ends developed by Hopf and Freudenthal. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]