The authors develop thorough and complete foundations for the method of almost etale extensions, which is at the basis of Faltings' approach to p-adic 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 V-Mod/S of the category V-Mod 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...