  We study the statistics of the largest eigenvalue lambda_max of N x N random
matrices with unit variance, but power-law distributed entries, P(M_{ij})~
|M_{ij}|^{-1-mu}. When mu > 4, lambda_max converges to 2 with Tracy-Widom
fluctuations of order N^{-2/3}. When mu < 4, lambda_max is of order
N^{2/mu-1/2} and is governed by Fr\'echet statistics. The marginal case mu=4
provides a new class of limiting distribution that we compute explicitely. We
extend these results to sample covariance matrices, and show that extreme
events may cause the largest eigenvalue to significantly exceed the
Marcenko-Pastur edge. Connections with Directed Polymers are briefly discussed.
