The deepest results of Abel, Riemann, Weierstrass, many of the most important papers of Klein and Poincare belong to this domain.
At the end of the last and the beginning of the present century the attitude towards algebraic geometry changed abruptly.
... Around the middle of the present century algebraic geometry had undergone to a large extent such a reshaping process.
The main task is to understand what kind of objects form "nice" families.
Once a good concept of "nice families" is established, the existence of a coarse moduli space should be nearly automatic.