  Using twisted nearby cycles, we define a new notion of slopes for complex
holonomic D-modules. We prove a boundedness result for these slopes, study
their functoriality and use them to characterize regularity. For a family of
(possibly irregular) algebraic connections E\_t parametrized by a smooth curve,
we deduce under natural conditions an explicit bound for the usual slopes of
the differential equation satisfied by the family of irregular periods of the
E\_t. This generalizes the regularity of the Gauss-Manin connection proved by
Katz and Deligne. Finally, we address some questions about analogues of the
above results for wild ramification in the arithmetic context.
