  The appearance of numerous period-doubling cascades is among the most
prominent features of {\bf parametrized maps}, that is, smooth one-parameter
families of maps $F:R \times {\mathfrak M} \to {\mathfrak M}$, where
${\mathfrak M}$ is a smooth locally compact manifold without boundary,
typically $R^N$. Each cascade has infinitely many period-doubling bifurcations,
and it is typical to observe -- such as in all the examples we investigate here
-- that whenever there are any cascades, there are infinitely many cascades. We
develop a general theory of cascades for generic $F$. We illustrate this theory
with several examples. We show that there is a close connection between the
transition through infinitely many cascades and the creation of a horseshoe.
