The first part of this paper is a review of the Strominger-Yau-Zaslow
conjecture in various settings. In particular, we summarize how, given a pair
(X,D) consisting of a Kahler manifold and an anticanonical divisor, families of
special Lagrangian tori in X-D and weighted counts of holomorphic discs in X
can be used to build a Landau-Ginzburg model mirror to X. In the second part we
turn to more speculative considerations about Calabi-Yau manifolds with
holomorphic involutions and their quotients. Namely, given a hypersurface H
representing twice the anticanonical class in a Kahler manifold X, we attempt
to relate special Lagrangian fibrations on X-H and on the (Calabi-Yau) double
cover of X branched along H; unfortunately, the implications for mirror
symmetry are far from clear.Comment: 27 pages, 1 figur
