The moduli space $\operatorname{Hom}(F,G)/G$ 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 $\operatorname{Hom}(F,G)/G$ and $\operatorname{Hom}(F,K)/K$ 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).