Appears in collection : New Structures and Techniques in $p$-adic Geometry

We will define the categories of (étale, rational) motives over an adic space $S$ and illustrate their most important properties, focusing on relevant applications in the study of $p$-adic cohomology theories. In particular, we will present the six-functor formalism they are equipped with, the continuity/spreading-out property, compact generation, and the identification between an analytic motive over a local field and a monodromy operator acting on its nearby cycle. We will sketch the proofs of these facts, highlighting the role of homotopies at each stage. Several applications will be presented, especially concerning the definition and study of rigid, de Rham, and Hyodo-Kato cohomologies.