We present some old and new results on a class of invariant spaces of holomorphic functions on symmetric domains, both in their circular bounded realizations and in their unbounded realizations as Siegel domains of type II. These spaces include: weighted Bergman spaces; the Hardy space \(H^2\) ; the Dirichlet space; holomorphic Besov spaces; the Bloch space. Our main focus will be on invariant Hilbert and semi-Hilbert spaces, but we shall also discuss minimal and maximal spaces in suitable classes of invariant Banach and semi-Banach spaces.