Jupiter's and Saturn's immense magnetospheres differ considerably from Earth's. These magnetospheres are generated in part by a strong planetary dynamo and by rapid rotation (~10 hour period). However, key differences lie in the internal sources of plasma (100s kg/s) provided by Io and Enceladus. Centrifugal stresses acting on the corotating, low-beta plasma in the inner magnetosphere lead to radial transport of plasma via a centrifugally-driven flux tube interchange instability. Instead of cooling on adiabatic expansion, the plasma is observed to be hotter at larger radial distances. In the outer magnetosphere the systems are governed by high-beta, centrifugally-confined plasma sheets. Observations and theories of the dynamics of Jupiter's and Saturn's magnetosphere will be discussed. In particular, we will focus on studies involving "viscous" processes (e.g. Kelvin-Helmholtz instability) at the magnetopause boundary that facilitate the transport of mass, momentum, and energy from the solar wind to the magnetosphere.