  Let $g_1, ..., g_k$ be tropical polynomials in $n$ variables with Newton
polytopes $P_1, ..., P_k$. We study combinatorial questions on the intersection
of the tropical hypersurfaces defined by $g_1, ..., g_k$, such as the
$f$-vector, the number of unbounded faces and (in case of a curve) the genus.
Our point of departure is Vigeland's work who considered the special case
$k=n-1$ and where all Newton polytopes are standard simplices. We generalize
these results to arbitrary $k$ and arbitrary Newton polytopes $P_1, ..., P_k$.
This provides new formulas for the number of faces and the genus in terms of
mixed volumes. By establishing some aspects of a mixed version of Ehrhart
theory we show that the genus of a tropical intersection curve equals the genus
of a toric intersection curve corresponding to the same Newton polytopes.
