Resumo: Given an oriented closed surface and an affine solvable Lie group G, the character variety is the moduli space of flat connections over a G-bundle over the surface. It is an affine symplectic variety with singularities. Character varieties first appeared in the work of Culler and Schalen on essential surfaces of 3-manifolds and attracted the attention of physicists and topologists for its relation with Chern-Simons topological quantum field theory and low dimensional topology. Geometers got involved in its study as well as these moduli spaces can be thought of as a generalization of the Teichmüller space of surfaces.