With D. Nadler.  Geometric Base Change.
We propose a geometric version of Langlands' base change principle. This describes the behavior of categories of sheaves on moduli of bundles under the passage from a curve to its covering spaces. As evidence for the conjecture we prove its spectral version (describing coherent sheaves on moduli of local systems on a curve and its coverings) and the special case of abelian groups. We then show how a twisted (or "orientifold") version of geometric base change for the covering P^1 ---> RP^2 implies Soergel's conjecture, a categorical form of the real local Langlands program. We also discuss other applications and relations with topological field theory.