  This is release 7.5 of our project, aiming to provide a complete treatment of
the foundations of almost ring theory, following and extending Faltings's
method of "almost etale extensions". The central result is the "almost purity
theorem", for whose proof we adapt Scholze's method, based on his perfectoid
spaces. This release provides the foundations for our generalization of
Scholze's perfectoid spaces, and reduces the proof of the almost purity theorem
to a general assertion concerning the \'etale topology of adic spaces, whose
proof uses previous work by the first author. As usual, this new release is a
mix of corrections and various improvements, with a final chapter dedicated to
applications; notably, we include a generalization of Y.Andr\'e's "perfectoid
Abhyankar's lemma" which we use to give a proof of a generalization of the
"direct summand conjecture", extending Andr\'e's recent work.
