  We give a rigorous definition of tropical fans (the "local building blocks
for tropical varieties") and their morphisms. For such a morphism of tropical
fans of the same dimension we show that the number of inverse images (counted
with suitable tropical multiplicities) of a point in the target does not depend
on the chosen point - a statement that can be viewed as the beginning of a
tropical intersection theory. As an application we consider the moduli spaces
of rational tropical curves (both abstract and in some R^r) together with the
evaluation and forgetful morphisms. Using our results this gives new, easy, and
unified proofs of various tropical independence statements, e.g. of the fact
that the numbers of rational tropical curves (in any R^r) through given points
are independent of the points.
