The dynamics of the vigorous convection in the outer envelope of the Sun must determine the transport of energy, angular momentum, and magnetic fields and must therefore be responsible for the observed surface activity and the angular velocity profile inferred helioseismically from SOI-MDI p-mode frequency splittings. Many different theoretical treatments have been applied to the problem, ranging from simple physical models such as mixing-length theory to sophisticated numerical simulations. Although mixing-length models provide a good first approximation to the structure of the convection zone, recent progress has mainly come from numerical simulations. Computational constraints have until now limited simulations in full spheres to essentially laminar convection. The angular velocity profiles have shown constancy on cylinders, in striking contrast to the approximately constant angular velocity on radial lines inferred for the Sun. In an effort to further our understanding of the dynamics of the solar convection zone, we have developed a new computer code that, by exploiting massively parallel architectures, enables us to study fully turbulent spherical shell convection. Here we present five fully evolved solutions. Motivated by the fact that a constant entropy upper boundary condition produces a latitudinal modulation of the emergent energy flux (of about 10%, i.e., far larger than is observed for the Sun), three of these solutions have a constant energy flux upper boundary condition. This leads to a latitudinal modulation of the specific entropy that breaks the constancy of the angular velocity on cylinders, making it more nearly constant on radial lines at midlatitudes. The effect of lowering the Prandtl number is also considered—highly time-dependent, vortical convective motions are revealed, and the Reynolds stresses are altered, leading to a reduced differential rotation. The differential rotation in all of our simulations shows a balance between driving by Reynolds stresses and damping by viscosity. This contrasts with the situation in the Sun, where the effect of viscosity on the mean differential rotation is almost negligible.