We describe a new approach to relative p -adic Hodge theory

based on systematic use of Witt vector constructions and nonar-chimedean

analytic geometry in the style of both Berkovich and

Huber. We give a thorough development of Phi-modules over a relative

Robba ring associated to a perfect Banach ring of characteristic

p , including the relationship between these objects and

étale Z_p-local systems and Q_p-local systems on the algebraic and

analytic spaces associated to the base ring, and the relationship

between (pro-)étale cohomology and Phi-cohomology. We also make

a critical link to mixed characteristic by exhibiting an equivalence

of tensor categories between the finite étale algebras over an arbitrary

perfect Banach algebra over a nontrivially normed complete

field of characteristic p and the finite étale algebras over a

corresponding Banach Q_p-algebra. This recovers the homeomorphism

between the absolute Galois groups of F_p((Pi)) and Q_p(µ_pInfini)

given by the field of norms construction of Fontaine and Wintenberger,

as well as generalizations considered by Andreatta, Brinon,

Faltings, Gabber, Ramero, Scholl, and most recently Scholze. Using

Huber's formalism of adic spaces and Scholze's formalism of

perfectoid spaces, we globalize the constructions to give several

descriptions of the étale local systems on analytic spaces over p -adic

fields. One of these descriptions uses a relative version of the

Fargues-Fontaine curve.