Fermions, particles with half-integer spin such as electrons, protons and neutrons, are the building blocks of matter. When fermions strongly interact, complex phenomena emerge, for example high-temperature superconductivity or superfluidity in neutron stars. Ultracold Fermi gases of atoms are a new type of strongly interacting fermionic matter that can be created and studied in the laboratory with exquisite control. For example, we can study the collision of "spin up" and "spin down" Fermi gases with the strongest interactions allowed by quantum mechanics. In equilibrium, direct absorption images of the trapped atomic gas reveal the entire thermodynamics of the system, including the transition into the superfluid state. Scaled to the density of electrons, superfluidity would occur far above room temperature. We were recently able to follow the evolution of fermion pairing from three to two dimensions, connecting quite directly to models of layered superconductors. Our measurements in and out of equilibrium provide benchmarks for current many-body theories and will help to understand other strongly interacting Fermi systems, such as high-temperature superconductors and neutron stars.