  Let $X$ be a compact connected Riemann surface of genus at least two, and let
${G}$ be a connected semisimple affine algebraic group defined over $\mathbb
C$. For any $\delta \in \pi_1({G})$, we prove that the moduli space of
semistable principal ${G}$--bundles over $X$ of topological type $\delta$ is
simply connected. In contrast, the fundamental group of the moduli stack of
principal ${G}$--bundles over $X$ of topological type $\delta$ is shown to be
isomorphic to $H^1(X, \pi_1({G}))$.
