Appears in collection : Conférence de lancement de la Chaire Jean-Pierre Bourguignon

I would like to explain in this talk how questions in non-abelian p-adic Hodge theory and in the theory of locally analytic representations of p-adic groups lead to consider new geometric objects attached to rigid analytic spaces, which require to go beyond the formalism of diamonds and are naturally defined in the analytic geometry framework developed by Clausen-Scholze. Based on a joint project (very much in progress) with Anschütz, Rodriguez Camargo and Scholze.