Given an algebraic curve $f(x,y)\in \mathbb{C}[x,y]$ of genus $g$, we have the following well-known result:
- If $g=0$, the curve can be parametrized by rational functions $x(t),y(t)\in \mathbb{C}(t)$.
- If $g=1$, the curve can be parametrized by elliptic functions $x(t),y(t)\in \mathbb{C}(\wp(t),\wp'(t))$.
- If $g>1$, the curve can be parametrized by suitable automorphic functions.
What is known about the analogous problem for complex algebraic surfaces? For rational or ruled surfaces the answer is obvious, but what is known about parametrizing surfaces of other types?