  This paper studies the derived de Rham cohomology of F_p and p-adic schemes,
and is inspired by Beilinson's recent work. Generalising work of Illusie, we
construct a natural isomorphism between derived de Rham cohomology and
crystalline cohomology for lci maps of such schemes, as well logarithmic
variants. These comparisons give derived de Rham descriptions of the usual
period rings and related maps in p-adic Hodge theory. Placing these ideas in
the skeleton of Beilinson's construction leads to a new proof of Fontaine's
crystalline conjecture and Fontaine-Jannsen's semistable conjecture.
