  Let $\Gamma$ be a finitely generated group, $G_\mathbb{C}$ be a classical
complex group and $G_\mathbb{R}$ a real form of $G_\mathbb{C}$. We propose a
definition of the $G_\mathbb{R}$-character variety of $\Gamma$ as a subset
$\mathcal{X}_{G_\mathbb{R}}(\Gamma)$ of the $G_\mathbb{C}$-character variety
$\mathcal{X}_{G_\mathbb{C}}(\Gamma)$. We prove that these subsets cover the set
of irreducible $G_\mathbb{C}$-characters fixed by an anti-holomorphic
involution $\Phi$ of $\mathcal{X}_{G_\mathbb{C}}(\Gamma)$. Whenever
$G_\mathbb{R}$ is compact, we prove that $\mathcal{X}_{G_\mathbb{R}}(\Gamma)$
is homeomorphic to the topological quotient
$\mathrm{Hom}(\Gamma,G_\mathbb{R})/G_\mathbb{R}$. Finally, we identify the
reducible points of $\mathcal{X}_{\mathrm{GL}(n,\mathbb{C})}(\Gamma)$ fixed by
an anti-holomorphic involution $\Phi$ as coming from direct sums of
representations with values in real groups.
