The great references given on <a href="http://mathoverflow.net/questions/10860/why-no-abelian-varieties-over-z"><I>Ilya's question</I></a> make me wonder about the current status of the many conjectures and open questions in <a href="http://www.ams.org/mathscinet-getitem?mr=1265522"><I>Illusie's survey</I></a> from 1994 on <a href="http://en.wikipedia.org/wiki/Crystalline_cohomology"><I>crystalline cohomology</I></a>. Obviously (just compare Illusie's survey from 1975 with that above or with <a href="http://perso.univ-rennes1.fr/antoine.chambert-loir/publications/papers/cristal.ps.gz"><I>Chambert-Loir's survey</I></a> from 1998), there is very intense work on that and the connections between the various cohomology theories attacking the case "l=p". Some more recent surveys only on Fontaine's p-adic Hodge theory are already linked to in the answers to Ilya's question, <a href="http://www.maa.org/maa%20reviews/RC.html"><I>Le Stum's book</I></a> (<a href="http://perso.univ-rennes1.fr/bernard.le-stum/Documents/RC-Errata.pdf"><I>Errata</I></a>) covers rigid chohomology. Among the open issues mentioned in Illusie's survey are finiteness theorems, crystalline coefficients, geometric semistability, the identity of characteristic polynomials of the Frobenius of different theories,... What is the current status of these? Which new theories have been created the past decade, how fit they together and which new questions emerged?   

Edit: U. Jannsen talked <a href="http://www.math.uvsq.fr/galois2010/resumsemaineFont/Jannsen.html"><I>recently</I></a> on "a refinement of crystalline cohomology by using the theory of so-called gauges as introduced earlier by Mazur and Kato and certain syntomic sheaves." Unfortunately I found no preprint on that.