  We introduce the notion of a prism, which may be regarded as a "deperfection"
of the notion of a perfectoid ring. Using prisms, we attach a ringed site --
the prismatic site -- to a $p$-adic formal scheme. The resulting cohomology
theory specializes to (and often refines) most known integral $p$-adic
cohomology theories.
  As applications, we prove an improved version of the almost purity theorem
allowing ramification along arbitrary closed subsets (without using adic
spaces), give a co-ordinate free description of $q$-de Rham cohomology as
conjectured by the second author, and settle a vanishing conjecture for the
$p$-adic Tate twists $\mathbf{Z}_p(n)$ introduced in previous joint work with
Morrow.
