  We report on the problem of the existence of complex and real algebraic
curves in the plane with prescribed singularities up to analytic and
topological equivalence. The question is whether, for a given positive integer
$d$ and a finite number of given analytic or topological singularity types,
there exist a plane (irreducible) curve of degree $d$ having singular points of
the given type as its only singularities. The set of all such curves is a
quasi-projective variety, which we call an equisingular family (ESF). We
describe, in terms of numerical invariants of the curves and their
singularities, the state of the art concerning necessary and sufficient
conditions for the non-emptiness and $T$-smoothness (i.e., smooth of expected
dimension) of the corresponding ESF. The considered singularities can be
arbitrary, but we spend special attention to plane curves with nodes and cusps,
the most studied case, where still no complete answer is known in general. An
important result is, however, that the necessary and the sufficient conditions
show the same asymptotics for $T$-smooth equisingular families if the degree
goes to infinity.
