Groups are omnipresent in mathematics. They appear naturally when describing symmetries of various mathematical objects. Many of those groups come equipped with a natural topology which is compatible with the algebraic structure. Therefore, it makes sense to use this additional information when discussing the group (and its actions on other objects). The goal of this lecture is to provide an overview in the theory of topological groups and to give many interesting examples. Besides other topics, we explore the interaction of topological properties (like connectedness, compactness and metrizability) with the group structure. Furthermore, we will explore when a continuous group homomorphism is automatically open (Open Mapping Theorems for topological groups). Differential structures on groups (Lie groups) will not be a topic of this course, although there are plenty of connections between the corresponding theories.