We will give an introduction to the theory of compact Lie groups and their representations, following the book Representations of compact Lie groups by Bröcker and tom Dieck. Particular theorems we will discuss are the Peter-Weyl theorem, which implies that every compact Lie group is a subgroup of a matrix group, and the Conjugation Theorem, which says that every Lie group admits up to conjugation a unique maximal torus. Finally, we will discuss the classification of connected compact Lie groups in terms of root systems.