In this course serves as a first introduction to the mathematical aspects of topological field theories, focusing on low-dimensional cases, especially dimension 3. We will briefly go through basic notions in category theory to give the definition of topological field theory, then we will explore the classification of topological field theories in dimensions 1 and 2. For higher dimensional TFTs, we start with Dijkgraaf-Witten theory, and then focus particularly on dimension 3. We will explain how to construct a 3d-TFT with anomaly from a modular fusion category. We will then learn about its non-semisimple version if there is time.