Topics in the arithmetic of Del Pezzo and K3 surfaces

In this thesis we study the arithmetic of certain del Pezzo surfaces and K3 surfaces.We prove that all the del Pezzo surfaces of degree 2 over a finite field are unirational. We compute the Picard lattice of the members of a family of K3 surfaces given by double covers of the projective plane. Finally, we provide an explicit example of a K3 surface over the field of rational numbers with a particular Picard lattice of rank 2.