We have made VLA images of the fields around three young pulsars which have resulted in the discovery of two new supernova remnants and confirmation of a third. We argue that, in at least two cases and perhaps the third, the pulsars are physically associated with these supernova remnants. A review of all known young pulsars shows that the majority are associated with supernova remnants. We show that the typical density of the interstellar medium into which the supernova remnants are evolving has a density of 0.2 cm$^{-3}$ instead of the low value of 0.01 cm$^{-3}$ which had been calculated from other studies, and results in a considerably longer radio lifetime for supernova remnants. Both the morphology of the supernova remnants and the location of the pulsars imply that most of these young pulsars are born with large transverse velocities ($sim$500 km s$^{-1}$). This high velocity mean in the distribution of pulsar velocities appears to be a general property of the pulsar population at birth, not seen in proper motion studies, due to selection effects. We explore the implications of this result as it relates to the origin of these velocities and the galactic distribution of pulsars. High velocity pulsars can escape their supernova remnant in a very short timescale, comparable to the lifetime of the remnant and may even play a role in extending the observable radio lifetime of the remnant. A significant fraction will be capable of escaping the disk of the Galaxy, producing an extended halo population.