Given a tropical variety X and two non-negative integers p and q we define
a homology group Hp,q (X) which is a finite-dimensional vector space over Q. We
show that if X is a smooth tropical variety that can be represented as the tropical limit
of a 1-parameter family of complex projective varieties, then dim Hp,q (X) coincides
with the Hodge number h p,q of a general member of the family.