  We study the fluctuations of the largest eigenvalue $\lambda_{\max}$ of $N
\times N$ random matrices in the limit of large $N$. The main focus is on
Gaussian $\beta$-ensembles, including in particular the Gaussian orthogonal
($\beta=1$), unitary ($\beta=2$) and symplectic ($\beta = 4$) ensembles. The
probability density function (PDF) of $\lambda_{\max}$ consists, for large $N$,
of a central part described by Tracy-Widom distributions flanked, on both
sides, by two large deviations tails. While the central part characterizes the
typical fluctuations of $\lambda_{\max}$ -- of order ${\cal O}(N^{-2/3})$ --,
the large deviations tails are instead associated to extremely rare
fluctuations -- of order ${\cal O}(1)$. Here we review some recent developments
in the theory of these extremely rare events using a Coulomb gas approach. We
discuss in particular the third-order phase transition which separates the left
tail from the right tail, a transition akin to the so-called Gross-Witten-Wadia
phase transition found in 2-d lattice quantum chromodynamics. We also discuss
the occurrence of similar third-order transitions in various physical problems,
including non-intersecting Brownian motions, conductance fluctuations in
mesoscopic physics and entanglement in a bipartite system.
