  Shear viscosity is a measure of the amount of dissipation in a simple fluid.
In kinetic theory shear viscosity is related to the rate of momentum transport
by quasi-particles, and the uncertainty relation suggests that the ratio of
shear viscosity eta to entropy density s in units of hbar/k_B is bounded by a
constant. Here, hbar is Planck's constant and k_B is Boltzmann's constant. A
specific bound has been proposed on the basis of string theory where, for a
large class of theories, one can show that eta/s is greater or equal to hbar/(4
pi k_B). We will refer to a fluid that saturates the string theory bound as a
perfect fluid. In this review we summarize theoretical and experimental
information on the properties of the three main classes of quantum fluids that
are known to have values of eta/s that are smaller than hbar/k_B. These fluids
are strongly coupled Bose fluids, in particular liquid helium, strongly
correlated ultracold Fermi gases, and the quark gluon plasma. We discuss the
main theoretical approaches to transport properties of these fluids: kinetic
theory, numerical simulations based on linear response theory, and holographic
dualities. We also summarize the experimental situation, in particular with
regard to the observation of hydrodynamic behavior in ultracold Fermi gases and
the quark gluon plasma.
