  We give a tropical description of the counting of real log curves in toric
degenerations of toric varieties. We treat the case of genus zero curves and
all non-superabundant higher-genus situations. The proof relies on log
deformation theory and is a real version of the Nishinou-Siebert approach to
the tropical correspondence theorem for complex curves. In dimension two, we
use similar techniques to study the counting of real log curves with
Welschinger signs and we obtain a new proof of Mikhalkin's tropical
correspondence theorem for Welschinger invariants.
