An extensive series of laboratory experiments is used to investigate the behavior of sheared thermal plumes. The plumes are generated by heating a small circular plate on the base of a cylindrical tank filled with viscous fluid and then sheared by rotating a horizontal lid at the fluid surface. The motion of passive tracers in the plumes is visualized by the release of several dye streams on the hot plate. We systematically examine the dependence of the convective flow on four dimensionless numbers: a velocity ratio, a Rayleigh number, the viscosity ratio, and an aspect ratio. We identify and delineate two transitions in the convective behavior: from a regime where the plume can spread upstream against the shear to a regime where the entire plume is advected downstream, and from a regime of negligible cross-stream circulation to a regime with significant cross-stream circulation and thermal entrainment. Our analysis of the steady profiles of the plumes shows that they initially rise with a constant vertical rise velocity. This rise velocity depends on the buoyancy flux and ambient viscosity but is almost independent of the centerline plume viscosity, which suggests that most of the thermal plume has a viscosity that is much closer to the ambient viscosity than the centerline viscosity. As the plumes approach the lid, they decelerate as the viscous drag on them steadily increases. The lateral spreading of the plumes under the lid is found to be well described by similarity solutions derived for the spreading of compositional plumes on a rigid surface, if the effective viscosity of the thermal plumes is taken to be the ambient viscosity rather than the centerline viscosity. A similar theoretical model is found to roughly predict the upstream spreading of thermal plumes at low shear, but it breaks down at moderate to high shear, where the entire plumes are advected downstream. When our results are applied to the Earth, we find that mantle plumes are mostly divided into only two flow regimes in the upper mantle: plumes under slow moving plates experience upstream flow and negligible cross-stream circulation, while plumes under faster moving plates (including all Pacific plumes) experience significant cross-stream circulation and are advected downstream. We also demonstrate that geochemical heterogeneities in a plume's source region will result in an azimuthally zoned plume and in an asymmetric geographical distribution of geochemical heterogeneities in the erupted hot spot basalts, as is seen in the Hawaiian, Gal¿pagos, Marquesas, and Tahiti/Society island chains. For individual mantle plumes, we determine their diameter and vertical rise velocity as well as the extent of upstream spreading and the rate of lateral spreading under the lithosphere.