  The Ehrhart polynomial $\text{ehr}_P(n)$ of a lattice polytope $P$ counts the
number of integer points in the $n$-th integral dilate of $P$. The $f^*$-vector
of $P$, introduced by Felix Breuer in 2012, is the vector of coefficients of
$\text{ehr}_P(n)$ with respect to the binomial coefficient basis $
\left\{\binom{n-1}{0},\binom{n-1}{1},...,\binom{n-1}{d}\right\}$, where $d =
\dim P$. Similarly to $h/h^*$-vectors, the $f^*$-vector of $P$ coincides with
the $f$-vector of its unimodular triangulations (if they exist). We present
several inequalities that hold among the coefficients of $f^*$-vectors of
polytopes. These inequalities resemble striking similarities with existing
inequalities for the coefficients of $f$-vectors of simplicial polytopes; e.g.,
the first half of the $f^*$-coefficients increases and the last quarter
decreases. Even though $f^*$-vectors of polytopes are not always unimodal,
there are several families of polytopes that carry the unimodality property. We
also show that for any polytope with a given Ehrhart $h^*$-vector, there is a
polytope with the same $h^*$-vector whose $f^*$-vector is unimodal.
