Extensive molecular dynamics simulations are carried out to study static and dynamic properties of symmetric diblock copolymer melts, both in the disordered and in the lamellar phases. The lamellar phase is constructed using the natural lamellar spacing, determined from constant pressure simulations. The non-Gaussian character of the chains in the disordered phase is demonstrated and quantified. In the lamellar phase, the density profile of the separate blocks, as well as the interface thickness are determined as a function of chain length N and AB interaction parameter <(epsilon)over tilde>, and compared with experiments and other theoretical results. Single chain and single block form factors indicate that the chains in the lamellar phase are distorted into stick-like objects. Our results in the disordered phase show a stronger dependence of the diffusion constant on the chain length than observed in previous simulations. Diffusion within the lamellar plane for systems with chains of length N less than or equal to 100 monomers is found to be almost independent of <(epsilon)over tilde>, in agreement with the predictions by Barrat and Fredrickson for Rouse chains. Diffusion perpendicular to the lamellae is exponentially suppressed with increasing <(epsilon)over tilde>. Simulations with even longer chains (up to N = 400 monomers) indicate that, in the strong segregation regime, chain stretching lowers the entanglement density and shifts the tube motion characteristic of the chain dynamics in homogeneous melts of long chains toward much longer chains.