  Let $X$ be an irreducible smooth projective curve of genus $g \geq 2$ over
$\mathbb{C}$. Let $G$ be a connected reductive affine algebraic group over
$\mathbb{C}$. Let $\mathrm{M}_{G, {\rm Higgs}}^{\delta}$ be the moduli space of
semistable principal $G$--Higgs bundles on $X$ of topological type $\delta \in
\pi_1(G)$. In this article, we compute the fundamental group and Picard group
of $\mathrm{M}_{G, {\rm Higgs}}^{\delta}$.
