To understand Galois representations is one of the central goals of number theory. This book is concerned with the p-adic Galois representations of a p-adic number field. Around 1990 Fontaine devised a strategy to compare such p-adic Galois representations to the seemingly much simpler objects of (semi)linear algebra, the so-called etale (ϕ,Γ)-modules. We will give a detailed and basically self-contained introduction to this theory. One of its key technical features is the close connection between the absolute Galois groups of local number fields and those of local function fields in positive characteristic. Instead of Fontaine’s original method we will use the very recent theory of perfectoid fields and the tilting correspondence to establish this connection. In addition, we will work in the more general framework of Lubin–Tate extensions of local number fields. Therefore the book also contains an introduction to the Lubin– Tate formal groups and to the formalism of ramified Witt vectors.