Rigid cohomology is certain extension of crystalline cohomology to schemes which are not necessarily proper or smooth, introduced in
It is also extending from the affine case the Monsky-Washnitzer cohomology.
The corresponding theory of coefficients is given by overconvergent isocrystals? and more generally arithmetic D-modules.
Kiran S. Kedlaya, Rigid cohomology and its coefficients, slides, pdf, workshop: p-adic Geometry and Homotopy Theory, Loen, 2009
Kiran Kedlaya, lecture 20, pdf from a course on algebraic geometry
wikipedia rigid cohomology
Denis-Charles Cisinski, Frédéric Déglise, Mixed Weil cohomologies, arxiv/0712.3291
Bernard le Stum, Rigid cohomology, Cambridge Tracts in Mathematics, 172 (2007)
David Brown, Rigid cohomology for algebraic stacks, Virginia 2010, slides
Warning: rigid cohomology in contemporary sense should not be confused with the topic of 1977 MacKenzie’s paper Rigid cohomology for topological groupoids (pdf)
See also arithmetic D-modules.