Limits of an increasing sequence of Riemann surfaces

Let $M$ be a Riemann surface which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $\mathrm{\Omega }\subset \mathbb{C}$. We describe $M$ in terms of $\mathrm{\Omega }$ under various assumptions on the boundary components of $\mathrm{\Omega }$.