  We study the eigenvalues of the covariance matrix $\frac{1}{n}M^*M$ of a
large rectangular matrix $M=M_{n,p}=(\zeta_{ij})_{1\leq i\leq p;1\leq j\leq n}$
whose entries are i.i.d. random variables of mean zero, variance one, and
having finite $C_0$th moment for some sufficiently large constant $C_0$. The
main result of this paper is a Four Moment theorem for i.i.d. covariance
matrices (analogous to the Four Moment theorem for Wigner matrices established
by the authors in [Acta Math. (2011) Random matrices: Universality of local
eigenvalue statistics] (see also [Comm. Math. Phys. 298 (2010) 549--572])). We
can use this theorem together with existing results to establish universality
of local statistics of eigenvalues under mild conditions. As a byproduct of our
arguments, we also extend our previous results on random Hermitian matrices to
the case in which the entries have finite $C_0$th moment rather than
exponential decay.
