  These are lecture notes prepared for the summer school "Geometric, algebraic
and topological methods in quantum field theory", held in Villa de Leyva in
July 2017. Our goal is to provide an introduction to a conjecture of Chern that
states that the Euler characteristic of a closed affine manifold vanishes. We
present part of the history and motivation for the conjecture as well as some
recent developments. All comments and corrections are most welcome!
