We define and study the arithmetic de Rham stack Xarith of a scheme X over a field of characteristic p, and analyze its relation with related stacks such as the Hyodo--Kato stack. We show that Xarith
Nearby in the stack
gives a stack-theoretic approach to rigid cohomology and its coefficients, known as overconvergent isocrystals and arithmetic
D
-modules, which avoids the long-standing problem of frame-choosing in earlier approaches. We finally give some arithmetic applications of our formalism, such as a proof of Berthelot's conjecture on the preservation of overconvergent isocrystals by smooth proper pushforward.