The moduli space $Hom(F,G)/G$ of completely reducible representations of a
finitely generated group $F$ into a Lie group $G$, known as the $G$-character
variety of $F$, appears naturally in connection with knot theory, Higgs
bundles and quantum field theories.
We will discuss the geometry, topology and singularities of these varieties
in the case when $G$ is a complex affine reductive Lie group with maximal
compact subgroup $K$, and $F$ is a free group of rank $r$. In this situation,
one can show that $Hom(F,G)/G$ and $Hom(F,K)/K$ have the same homotopy type.
Moreover, if $G=SL(n,C)$, these character varieties admit a smooth structure
only when $F$ or $G$ is abelian, or $r+n\leq 5$. Moreover, in the cases when
$r+n=5$, these moduli spaces have the homotopy type of spheres. This is joint
work with S. Lawton (arxiv:0807.3317 and arXiv:0907.4720).