When vegetative bacteria that can swim are grown in a rich medium on an agar surface, they become multinucleate, elongate, synthesize large numbers of flagella, produce wetting agents, and move across the surface in coordinated packs: they swarm. We examined the motion of swarming Escherichia coli, comparing the motion of individual cells to their motion during swimming. Swarming cells' speeds are comparable to bulk swimming speeds, but very broadly distributed. Their speeds and orientations are correlated over a short distance (several cell lengths), but this correlation is not isotropic. We observe the swirling that is conspicuous in many swarming systems, probably due to increasingly long-lived correlations among cells that associate into groups. The normal run-tumble behavior seen in swimming chemotaxis is largely suppressed, instead, cells are continually reoriented by random jostling by their neighbors, randomizing their directions in a few tenths of a second. At the edge of the swarm, cells often pause, then swim back toward the center of the swarm or along its edge. Local alignment among cells, a necessary condition of many flocking theories, is accomplished by cell body collisions and/or short-range hydrodynamic interactions.