In this talk we introduce a new notion of semismoothness
which pertains both sets as well as multifunctions. In the case
of single-valued maps it is closely related with the standard
notion of semismoothness introduced by Qi and Sun in 1993.
Semismoothness can be equivalently characterized in terms of
regular, limiting and directional limiting coderivatives.
Then we present a semismooth* Newton method for solving inclusions and
generalized equations (GEs), where the linearization concerns
both the single-valued and the multi-valued part and it is
performed on the basis of the respective coderivatives. Two
conceptual algorithms will be presented and shown to converge
locally superlinearly under relatively weak assumptions.