  In this paper we first show that each Kummer quartic surface (a quartic
surface $X$ with 16 singular points) is, in canonical coordinates, equal to its
dual surface, and that the Gauss map induces a fixpoint free involution
$\gamma$ on the minimal resolution $S$ of $X$. Then we study the corresponding
Enriques surfaces $S/ \gamma$. We also describe in detail the remarkable
properties of the most symmetric Kummer quartic, which we call the Cefal\'u
quartic. We also investigate the Kummer quartic surfaces whose associated
Abelian surface is isogenous to a product of elliptic curves with kernel
$(\mathbb{Z}/2)^2$, and show the existence of polarized nodal K3 surfaces $X$
of any degree $d=2k$ with the maximal number of nodes, such that $X$ and its
nodes are defined over $\mathbb{R}$. We take then as parameter space for Kummer
quartics an open set in $\mathbb{P}^3$, parametrizing nondegenerate $(16_6,
16_6)$-configurations, and compare with other parameter spaces. We end with
some remarks on normal cubic surfaces.
