Summary:  This book develops thorough and complete foundations for the method of almost etale extensions, which is at the basis of Faltings' approach to padic Hodge theory. The central notion is that of an "almost ring". Almost rings are the commutative unitary monoids in a tensor category obtained as a quotient VMod/S of the category VMod of modules over a fixed ring V; the subcategory S consists of all modules annihilated by a fixed ideal m of V, satisfying certain natural conditions. The reader is assumed to be familiar with general categorical notions, some basic commutative algebra and some advanced homological algebra (derived categories, simplicial methods). Apart from these general prerequisites, the text is as selfcontained as possible. One novel feature of the book  compared with Faltings' earlier treatment  is the systematic exploitation of the cotangent complex, especially for the study of deformations of almost algebras
