  We introduce the fundamental group ${\mathcal F}(\mathcal{R}_{G, \phi})$ of a
uniquely ergodic Cantor minimal $G$-system $\mathcal{R}_{G, \phi}$ where $G$ is
a countable discrete group. We compute fundamental groups of several uniquely
ergodic Cantor minimal $G$-systems. We show that if $\mathcal{R}_{G, \phi}$
arises from a free action $\phi$ of a finitely generated abelian group, then
there exists a unital countable subring $R$ of $\mathbb{R}$ such that
$\mathcal{F}(\mathcal{R}_{G, \phi})=R_{+}^\times$. We also consider the
relation between fundamental groups of uniquely ergodic Cantor minimal
$\mathbb{Z}^n$-systems and fundamental groups of crossed product $C^*$-algebras
$C(X)\rtimes_{\phi} \mathbb{Z}^n$.
