  In this paper we prove some results about K3 surfaces with Picard number 1
and 2. In particular, we give a new simple proof of a theorem due to Oguiso
which shows that, given an integer $N$, there is a K3 surface with Picard
number 2 and at least $N$ non-isomorphic FM-partners. We describe also the
Mukai vectors of the moduli spaces associated to the Fourier-Mukai partners of
K3 surfaces with Picard number 1.
