  We propose a formulation of the Penrose plane wave limit in terms of null
Fermi coordinates. This provides a physically intuitive (Fermi coordinates are
direct measures of geodesic distance in space-time) and manifestly covariant
description of the expansion around the plane wave metric in terms of
components of the curvature tensor of the original metric, and generalises the
covariant description of the lowest order Penrose limit metric itself, obtained
in hep-th/0312029. We describe in some detail the construction of null Fermi
coordinates and the corresponding expansion of the metric, and then study
various aspects of the higher order corrections to the Penrose limit. In
particular, we observe that in general the first-order corrected metric is such
that it admits a light-cone gauge description in string theory. We also
establish a formal analogue of the Weyl tensor peeling theorem for the Penrose
limit expansion in any dimension, and we give a simple derivation of the
leading (quadratic) corrections to the Penrose limit of AdS_5 x S^5.
