  The four moment theorem asserts, roughly speaking, that the joint
distribution of a small number of eigenvalues of a Wigner random matrix (when
measured at the scale of the mean eigenvalue spacing) depends only on the first
four moments of the entries of the matrix. In this paper, we extend the four
moment theorem to also cover the coefficients of the \emph{eigenvectors} of a
Wigner random matrix. A similar result (with different hypotheses) has been
proved recently by Knowles and Yin, using a different method.
  As an application, we prove some central limit theorems for these
eigenvectors. In another application, we prove a universality result for the
resolvent, up to the real axis. This implies universality of the inverse
matrix.
