The geometry of impulsive pp-waves is explored via the analysis of the
geodesic and geodesic deviation equation using the distributional form of the
metric. The geodesic equation involves formally ill-defined products of
distributions due to the nonlinearity of the equations and the presence of the
Dirac \de-distribution in the space time metric. Thus, strictly speaking, it
cannot be treated within Schwartz's linear theory of distributions. To cope
with this difficulty we proceed by first regularizing the
\de-singularity,then solving the regularized equation within classical smooth
functions and, finally, obtaining a distributional limit as solution to the
original problem. Furthermore it is shown that this limit is independent of the
regularization without requiring any additional condition, thereby confirming
earlier results in a mathematical rigorous fashion. We also treat the Jacobi
equation which, despite being linear in the deviation vector field, involves
even more delicate singular expressions, like the ``square'' of the Dirac
\de-distribution. Again the same regularization procedure provides us with a
perfectly well behaved smooth regularization and a regularization-independent
distributional limit. Hence it is concluded that the geometry of impulsive
pp-waves can be described consistently using distributions as long as careful
regularization procedures are used to handle the ill-defined products.