  In the first part of this paper we give a survey of classical results on
Kummer surfaces with Picard number 17 from the point of view of lattice theory.
We prove ampleness properties for certain divisors on Kummer surfaces and we
use them to describe projective models of Kummer surfaces of (1,d)-polarized
Abelian surfaces for d=1,2,3. As a consequence we prove that in these cases the
N\'eron--Severi group can be generated by lines. In the second part of the
paper we use Kummer surfaces to obtain results on K3 surfaces with a symplectic
action of the group (\Z/2\Z)^4. In particular we describe the possible
N\'eron--Severi groups of the latter in the case that the Picard number is 16,
which is the minimal possible. We describe also the N\'eron--Severi groups of
the minimal resolution of the quotient surfaces which have 15 nodes. We extend
certain classical results on Kummer surfaces to these families.
