We provide the complete set of global solutions of viscous transonic flows (VTFs) around black holes and neutron stars. These solutions describe the optically thick and optically thin flows from the horizon of the black hole or from the neutron star surface to the location where the flow joins with a Keplerian disk. We study the nature of the multiple sonic points as functions of advection, rotation, viscosity, heating and cooling. Stable shock waves, which join two transonic solutions, are found to be present in a large region of the parameter space. We classify the solutions in terms of whether or not the flow can have a standing shock wave. We find no new topology of solutions other than what are observed in our previous studies of isothermal VTFs. We particularly stress the importance of the boundary conditions and argue that we have the most complete solution of accretion and winds around black holes and neutron stars.Comment: aasms style used. 49 pages of text and 9 figures; to appear in Astrophysical Journal, v. 464, June 20th, 199