Abstract: The tropical algebraic geometry, i.e. algebraic geometry over tropical (idempotent) semi-rings, has recently found fascinating applications to the enumeration of complex and real algebraic curves, and computing Gromov-Witten invariants of toric surfaces.

The idea of the tropical approach to these enumerative problems (Kontsevich, Mikhalkin) consists in reducing the count of algebraic curves to the count of tropical curves (non-Archimedean amoebas), which are just planar graphs dual to lattice subdivisions of Newton polygons.

We give an explanation of the link between the tropical and classical geometry, based on the presentation of a tropical object as limit (dequantization) of a classical object, and vice versa, a classical object as deformation (quantization) of a tropical object.

We shall discuss an application of the tropical approach to enumeration of real rational curves and computation of the Welschinger invariant (the Gromov-Witten invariant over the reals).