A mosaic of IRAS co-added images covering 8.7-degrees x 4.3-degrees reveals a large emission ''cavity'' with bright rims that extends some 5-degrees eastward from the Pleiades. This is a novel interstellar structure, unrelated to outflows or ionizing radiation from high-mass stars. Instead, it is a region pressurized by the soft-ultraviolet radiation of the cluster, most likely through photoelectric heating of the ambient interstellar medium. The IRAS emission cavity delineates the wake of the Pleiades as it moves supersonically through the interstellar medium. Evidence for the dynamical interaction that produces the wake exists in the morphology of the gas and dust, the orientation of interstellar polarization, and especially in the velocities of the atomic and molecular gas. The magnitude of radial velocity shifts near the cluster provide constraints on the orientation of the gas flow, and hence on the space motion of the gas. These data suggest that the interaction involves generation of a shock wave. Consideration of possible mechanisms for the cluster-cloud interaction identifies photoelectric heating as the most likely agent and leads to the suggestion that transverse expansion of heated gas near the cluster plays a crucial role in driving the shock. The analysis indicates that the interstellar matter in the Pleiades is undergoing a chance collision with the cluster at approximately 18 km s-1; it is unrelated to the birth of the cluster. The wake orientation offers an alternative and independent approach to the cloud motion. Tracing the cloud trajectory back to an origin in Gould's Belt with transverse velocity as a free parameter yields a less precise but virtually identical result for the space motion as the analysis of the kinematics. The orbit reconstruction suggests that the cloud originated approximately 15 Myr ago in an energetic event near l = 60-degrees and low positive b. The present-day position of the hypothesized source exhibits a local minimum in molecular gas and a 21 cm morphology that suggests a blow-out of gas into the Galactic halo, but no recognized star cluster or stellar association exists there. The pulsar PSR 1919+21, which has a spin-down age of 15 Myr but no detected proper motion, lies tantalizingly nearby. The orientation of the Pleiades wake by itself implies that this material is kinematically unrelated to the star-forming complexes in Taurus and Perseus, in spite of proximity in space and similarity in radial velocity. Review of data for the larger Taurus/Perseus region leads to a schematic view of the spatial structure of the interstellar medium in this region, and to some final remarks on the evolution of the local system of gas and young stars.