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).