Under what conditions do there exist non-constant meromorphic functions between general Riemann surfaces?

Question

The uniformization theorem answers this question for particular Riemann surfaces, but do we have a general theorem for this? Do we also get meromorphic functions that can separate points?

Answer 1

Given a complex Riemann surface $X$ and two points $x,y\in X$ there always exist a meromorphic function $f\in \mathcal M (X)$ which is holomorphic at both $x$ and $y$ and such that $f(x)\neq f(y)\in \mathbb C$.
a) If $X$ is non-compact you may even assume that $f\in \mathcal O(X)$ is holomorphic everywhere.
b) But if $X$ is compact then $\mathcal O(X)=\mathbb C$ and $f$ will necessarily have at least one pole .

These are highly non-trivial results going back to Riemann but of course his proofs do not meet contemporary standards of rigor: the first precise definition of "Riemann surface" goes back to Weyl's Die Idee der Riemannschen Fläche, published in 1913, 47 years after Riemann's death in 1866...
A high-brow explanation for these results is that a non compact Riemann surface is a Stein manifold (Behnke-Stein 1948) and that a compact Riemann surface is a projective manifold, to which Serre's GAGA principle applies.
By far the best modern reference for Riemann surfaces is Otto Forster's book.