Let $G/K$ be a Riemannian symmetric space. Its complexification $G^C / K^C$ is a Stein manifold, and left-translations by $G$ are holomorphic transformations of $G^C / K^C$. In this setting, invariant domains and their envelopes of holomorphy are natural objects of study. If $G/K$ is compact, then every invariant domain $D$ in $G^C / K^C$ intersects a complex torus orbit in a lower dimensional Reinhardt domain $\Omega_D$. In this case, complex analytic properties of $D$ can be expressed in terms of those of $\Omega_D$. If $G/K$ is a non-compact, then the situation is fully understood only in the rank-one case. In this talk we present some univalence results for the envelope of holomorphy of a $G$-invariant domain in $G^C / K^C$, when the space $G/K$ is a non-compact Hermitian symmetric space (joint work with A. Iannuzzi).