  The Unitary Fermi Gas (UFG) is one of the most strongly interacting systems
known to date, as it saturates the unitarity bound on the quantum mechanical
scattering cross section. The UFG corresponds to a two-component Fermi gas in
the limit of short interaction range and large scattering length, and is
currently realized in ultracold-atom experiments via Feshbach resonances. While
easy to define, the UFG poses a challenging quantum many-body problem, as it
lacks any characteristic scale other than the density. As a consequence,
accurate quantitative predictions of the thermodynamic properties of the UFG
require Monte Carlo calculations. However, significant progress has also been
made with purely analytical methods. Notably, in 2005 Tan derived a set of
exact thermodynamic relations in which a universal quantity known as the
"contact" C plays a crucial role. Recently, C has also been found to determine
the prefactor of the high- frequency power-law decay of correlators as well as
the right-hand-sides of shear- and bulk viscosity sum rules. The contact is
therefore a central piece of information on the UFG in equilibrium as well as
away from equilibrium. In this talk we describe some of the known aspects of
Fermi gases at and around unitarity, show our latest Monte Carlo results for
the contact at finite temperature, and summarize the open questions in the
field, some of which we are starting to answer using large-scale Monte Carlo
calculations by adapting methods from Lattice QCD.
