  We study the tropicalizations of Severi varieties, which we call tropical
Severi varieties. In this paper, we give a partial answer to the following
question, ``describe the tropical Severi varieties explicitly.'' We obtain a
description of tropical Severi varieties in terms of regular subdivisions of
polygons. As an intermediate step, we construct explicit parameter spaces of
curves. These parameter spaces are much simpler objects than the corresponding
Severi variety and they are closely related to flat degenerations of the Severi
variety, which in turn describes the tropical Severi variety. As an
application, we understand G.Mikhalkin's correspondence theorem for the degrees
of Severi varieties in terms of tropical intersection theory. In particular,
this provides a proof of the independence of point-configurations in the
enumeration of tropical nodal curves.
