  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 through an isogeny
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 also extend to positive characteristic some results which were previously
known over $\mathbb{C}$.
  We end with a section devoted to remarks on normal cubic surfaces, and
providing some other examples of strictly selfdual hypersurfaces.
