  We review recent advances in the theory of trapped fermions using techniques
borrowed from random matrix theory (RMT) and, more generally, from the theory
of determinantal point processes. In the presence of a trap, and in the limit
of a large number of fermions $N \gg 1$, the spatial density exhibits an edge,
beyond which it vanishes. While the spatial correlations far from the edge,
i.e. close to the center of the trap, are well described by standard many-body
techniques, such as the local density approximation (LDA), these methods fail
to describe the fluctuations close to the edge of the Fermi gas, where the
density is very small and the fluctuations are thus enhanced. It turns out that
RMT and determinantal point processes offer a powerful toolbox to study these
edge properties in great detail. Here we discuss the principal edge
universality classes, that have been recently identified using these modern
tools. In dimension $d=1$ and at zero temperature $T=0$, these universality
classes are in one-to-one correspondence with the standard universality classes
found in the classical unitary random matrix ensembles: soft edge (described by
the "Airy kernel") and hard edge (described by the "Bessel kernel")
universality classes. We further discuss extensions of these results to higher
dimensions $d\geq 2$ and to finite temperature. Finally, we discuss
correlations in the phase space, i.e., in the space of positions and momenta,
characterized by the so called Wigner function.
