  Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It
allows for the computation of several cohomological invariants of an algebraic
variety. In particular, its application to enumerative algebraic geometry led
to significant progress.
  In this survey, we give an introduction to tropical geometry techniques for
algebraic curve counting problems. We also survey some recent developments,
with a particular emphasis on the computation of the degree of the Severi
varieties of the complex projective plane and other toric surfaces as well as
Hurwitz numbers and applications to real enumerative geometry. This paper is
based on the author's lecture at the Workshop on Tropical Geometry and
Integrable Systems in Glasgow, July 2011.
