  We develop the theory of semisimplifications of tensor categories defined by
Barrett and Westbury. In particular, we compute the semisimplification of the
category of representations of a finite group in characteristic $p$ in terms of
representations of the normnalizer of its Sylow $p$-subgroup. This allows us to
compute the semisimplification of the representation category of the symmetric
group $S_{n+p}$ in characteristic $p$, where $0\le n\le p-1$, and of the
Deligne category $\underline{\rm Rep}^{\rm ab}S_t$, where $t\in \Bbb N$. We
also compute the semisimplification of the category of representations of the
Kac-De Concini quantum group of the Borel subalgebra of $\mathfrak{sl}_2$. We
also study tensor functors between Verlinde categories of semisimple algebraic
groups arising from the semisimplification construction, and objects of finite
type in categories of modular representations of finite groups (i.e., objects
generating a fusion category in the semisimplification). Finally, we determine
the semisimplifications of the tilting categories of $GL(n)$, $SL(n)$ and
$PGL(n)$ in characteristic $2$. In the appendix, we classify categorifications
of the Grothendieck ring of representations of $SO(3)$ and its truncations.
