This course gives an introduction to modular categories with an emphasize on the 3-dimensional Reshetikhin-Turaev (RT) TQFTs. We will start by introducing the notion of modular categories and a graphical calculus on them. We will see how this hybrid of algebra and topology enables us to define invariants of framed links and 3-manifolds. We will then give an axiomatic definition of TQFTs and discuss some general properties of them. Finally, we will focus on the 3-dimensional RT-TQFT associated to modular categories with examples involving groups and quantum groups. Time permitted, we will discuss the mapping class group representations of an RT-TQFT in depth.