Merging algebra and analysis, this text uses Lie-theoretic methods to develop a theory with wide applications in mathematics and physics. It aims to encourage the reader's understanding of Lie theory to evolve from beginner to expert. Topics include: the Cartan theory of complex semisimple Lie algebras; the Cartan-Weyl theory of the structure of representations of compact Lie groups; the classification of real semisimple Lie algebras; the structure theory of noncompact reductive Lie groups; and integration on reductive groups.