  This paper studies the extreme gaps between eigenvalues of random matrices.
We give the joint limiting law of the smallest gaps for Haar-distributed
unitary matrices and matrices from the Gaussian unitary ensemble. In
particular, the kth smallest gap, normalized by a factor $n^{-4/3}$, has a
limiting density proportional to $x^{3k-1}e^{-x^3}$. Concerning the largest
gaps, normalized by $n/\sqrt{\log n}$, they converge in ${\mathrm{L}}^p$ to a
constant for all $p>0$. These results are compared with the extreme gaps
between zeros of the Riemann zeta function.
