Let K be a complete discretely valued field with perfect residue field F of characteristic p>0. Let X be a smooth projective variety over K. We formulate the notion of totally degenerate reduction of such a variety and prove a basic structure theorem for its etale cohomology. Examples include Tate and Mumford curves, and more generally Drinfeld modular varieties. We then discuss analogues of the Hodge and Tate conjectures for such varieties. This is in part joint work with X. Xarles (Barcelona)