  We study Levinson type theorems for the family of Aharonov-Bohm models from
different perspectives. The first one is purely analytical involving the
explicit calculation of the wave-operators and allowing to determine precisely
the various contributions to the left hand side of Levinson's theorem, namely
those due to the scattering operator, the terms at 0-energy and at infinite
energy. The second one is based on non-commutative topology revealing the
topological nature of Levinson's theorem. We then include the parameters of the
family into the topological description obtaining a new type of Levinson's
theorem, a higher degree Levinson's theorem. In this context, the Chern number
of a bundle defined by a family of projections on bound states is explicitly
computed and related to the result of a 3-trace applied on the scattering part
of the model.
