  This article, the first in a series, analyzes the general theory of plane
wave spacetimes. Following Dmitri Aleekseevsky, these are defined as spacetimes
admitting a group of dilations leaving invariant a smooth curve. If this curve
is specified as part of the structure, the spacetime is termed a Penrose limit,
whose theory was developed first by Roger Penrose. The main result is that
every plane wave is a Rosen universe, a generalization of the smooth metrics of
Albert Einstein and Nathan Rosen, allowing for certain isolated co-ordinate
singularities; the latter are characterized. We conclude with an extended
example, using the techniques developed in the article to associate a vacuum
plane wave in four dimensions to any hyperbolic billiard trajectory.
