I will explain some recent progress on local shtukas in characteristic p.
It is a joint work in progress with Eteve, Gaitsgory and Genestier. The proof uses nearby cycles over general bases
(after Deligne, Laumon, Gabber, Orgogozo, Hansen, Scholze), which I will recall.
I will also explain the compatibilities with global shtukas and multiplicity conjectures.