In this talk, we will discuss some recent advances in the theory of motivic rigid analytic geometry. In particular, we show how to define and study a relative de Rham cohomology for adic spaces in mixed characteristic and how to extend this construction to the equi-characteristic p case (taking values on the relative Fargues-Fontaine curve). Finiteness results, and the relation to "tilting" and to classical cohomology theories are also discussed. As an application, we give a proof of the p-adic weight-monodromy conjecture for smooth projective hypersurfaces. These results are part of joint works with J. Ayoub and M. Gallauer, with A.-C. Le Bras and with F. Binda and H. Kato.
