Motivated by dreams of finding a geometric incarnation of the p-adic Langlands correspondence in the cohomology of local Shimura varieties, in recent years, there has been an increasing interest in the p-adic Hodge theory of rigid-analytic varieties that are not necessarily proper (e.g. Stein). In this talk, I will give an introduction to this subject, and I will survey recent advances made by Colmez, Dospinescu, and Nizioł. In particular, I will report on a comparison theorem describing the geometric p-adic pro-étale cohomology in terms of de Rham data. If time permits, I will explain how, thanks to the Condensed Mathematics developed by Clausen and Scholze, one can prove such comparison also in cases where the relevant cohomology groups are otherwise pathological as topological vector spaces.
