I give a survey on results about the geometry of equisingular families V(s1,...,sr) of algebraic curves C in a linear system |D| on a fixed smooth projective surface S with prescribed singularities of given singularity type s1,...,sr (the types can be analytic or topological types or given by the Newton diagram, ...). A classical example is the family of plane curves of given degree in P^2 with nodes and cusps. We are interested in conditions guaranteeing the existence (non-emptiness) reps. smoothness resp. irreducibility of V(s1,...,sr). These conditions should be
- numerical: relating numerical invariants of the surface S, the linear system |D|, and s1,...,sr, - universal: applicable to any C, D, S, and an arbitrary number and type of singularities, - asymptotically proper: asymptotically of the same growth as the known necessary conditions
Substantial progress, even for curves in P^2, has been made only in the last decade. We show asymptotically proper conditions for existence in general (which are even optimal for plane curves with nodes and cusps). We discuss further proper and in some cases optimal asymptotic bounds for the T-smoothness and irreducibility of equisingular families and report on open problems and conjectures. Most of the results are joint work with Christoph Lossen and Eugenii Shustin.