When is a topological space $X$ the disjoint union of orbits under a continuous $G$-action?

Question

For $G$ a topological group acting continuously on a space $X$, when can we write the space (topologically) as the disjoint union of orbits i.e., when is $$ X\cong\coprod_{O\in X/G}O $$ Further, for $x\in O$, when is the canonical bijection a homeomorphism $$ O\cong G/\mathrm{Stab}\left(x\right) $$ I am looking for the conditions on $X$ and $G$, as well as on the action.