This thesis deals with K3 surfaces and their moduli spaces. In the first part we identify a class of complex K3 surfaces, called of zero entropy, with a particularly simple (but infinite) automorphism group, naturally arising from complex dynamics. We provide a lattice-theoretical classification of their N\'eron-Severi lattices. In the second part we move to the study of the Kodaira dimension of the moduli spaces of elliptic K3 surfaces of Picard rank $3$. We show that almost all of them are of general type, by using the low-weight cusp form trick developed by Gritsenko, Hulek and Sankaran. Moreover, we prove that many of the remaining moduli spaces are unirational, by providing explicit projective models of the corresponding K3 surfaces. In the final part, we investigate the set of rational points on K3 and Enriques surfaces over number fields. We show that all Enriques surfaces over number fields satisfy (a weak version of) the potential Hilbert property, thus proving that, after a field extension, the rational points on their K3 cover are dense and do not come from finite covers.