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Let $k$ be an algebraically closed field of positive characteristic $p$ and let $\mathbb{F}$ be an algebraically closed field of characteristic $0$. In this talk we introduce diagonal $p$-permutation functors: consider the $\mathbb{F}$-linear category $\mathbb{F}pp_k^{\Delta}$ of finite groups, in which the set of morphisms from $G$ to $H$ is the $\mathbb{F}$-linear extension of the Grothendieck group of $p$-permutation $(kH, kG)$-bimodules with (twisted) diagonal vertices. We call the $\mathbb{F}$-linear functors from $\mathbb{F}pp_k^{\Delta}$ to $\mathbb{F}$-$\mathrm{Mod}$ as diagonal  $p$-permutation functors.

We first consider the diagonal $p$-permutation functor of the $p$-permutation ring. We then show that the category of diagonal $p$-permutation functors is semisimple and give a description of the evaluations of simple functors at finite groups. Finally, we introduce the diagonal $p$-permutation functors arising, in a natural way, from the blocks of finite groups and show that $p$-permutation equivalent blocks give rise to isomorphic functors. This is joint work with Serge Bouc.