We will introduce the notion of perfectoid spaces. The theory can be seen as a kind of rigid geometry of infinite type, and the most important feature is that the theories over (deeply ramified extensions of) Q_p and over F_p((t)) are equivalent, generalizing to the relative situation a theorem of Fontaine-Wintenberger, and also implying a strong form of Faltings's almost purity theorem. This method of changing the characteristic is then applied to deduce many cases of the weight-monodromy conjecture.