  Statistical properties of eigenvectors in non-Hermitian random matrix
ensembles are discussed, with an emphasis on correlations between left and
right eigenvectors. Two approaches are described. One is an exact calculation
for Ginibre's ensemble, in which each matrix element is an independent,
identically distributed Gaussian complex random variable. The other is a
simpler calculation using $N^{-1}$ as an expansion parameter, where $N$ is the
rank of the random matrix: this is applied to Girko's ensemble. Consequences of
eigenvector correlations which may be of physical importance in applications
are also discussed. It is shown that eigenvalues are much more sensitive to
perturbations than in the corresponding Hermitian random matrix ensembles. It
is also shown that, in problems with time-evolution governed by a non-
Hermitian random matrix, transients are controlled by eigenvector correlations.
