  Given a simple graph $G$ on $n$ vertices, a subset of vertices $U \subseteq
V(G)$ is dominating if every vertex of $V(G)$ is either in $U$ or adjacent to a
vertex of $U$. The domination polynomial of $G$ is the generating function
whose coefficients are the number of dominating sets of a given size. We show
that the domination polynomial is unimodal, i.e., the coefficients are
non-decreasing and then non-increasing, for several well-known families of
graphs. In particular, we prove unimodality for spider graphs with at most
$400$ legs (of arbitrary length), lollipop graphs, arbitrary direct products of
complete graphs, and Cartesian products of two complete graphs. We show that
for every graph, a portion of the coefficients are non-increasing, where the
size of the portion depends on the upper domination number, and in certain
cases this is sufficient to prove unimodality. Furthermore, we study graphs
with $m$ universal vertices, i.e., vertices adjacent to every other vertex, and
show that the last $(\frac{1}{2} - \frac{1}{2^{m+1}}) n$ coefficients of their
domination polynomial are non-increasing.
