Notes on Topological Quantum Field Theories

These notes are the outcome of a mini-course on TQFTs held at the edition of Winter Braids in Pau in February 2015. We define the notion of TQFT and provide the first basic examples obtained via the universal construction and via Frobenius algebras. After recalling some basic notions on the mapping class groups of surfaces, we concentrate on the Reshetikhin-Turaev construction via the skein theoretical approach: we first define the skein module of a 3-manifold and the RT invariants; then we apply the universal construction to get the RT SU(2)-TQFTs. We conclude with an overview of the main results on these TQFTs and on some recent developments. An appendix summarizes the basic notions and facts in category theory used here.