in this talk, we will discuss some of the fundamental theorems of lie theory. we will discuss how to associate to a lie group (i.e. a smooth manifold that is also a group) a linear space, known as its lie algebra. we will then discuss how we can use lie algebras to classify different lie groups. we will discuss the fundamental theorem of sophus lie which tells us exactly when two lie groups are locally isomorphic, as well as the equivalence of the categories of real simply connected lie groups and finite dimensional real lie algebras. if time permits, we will discuss the study of real compact groups and their classification by certain types of complex lie algebras. we will also discuss algebraic groups (groups that are algebraic varieties) and algebraic lie algebras.