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.