  Starting with a brief introduction into the basics of relativistic fluid
dynamics, I discuss our current knowledge of a relativistic theory of fluid
dynamics in the presence of (mostly shear) viscosity. Derivations based on the
generalized second law of thermodynamics, kinetic theory, and a complete
second-order gradient expansion are reviewed. The resulting fluid dynamic
equations are shown to be consistent for all these derivations, when properly
accounting for the respective region of applicability, and can be applied to
both weakly and strongly coupled systems. In its modern formulation,
relativistic viscous hydrodynamics can directly be solved numerically. This has
been useful for the problem of ultrarelativistic heavy-ion collisions, and I
will review the setup and results of a hydrodynamic description of experimental
data for this case.
