Thurston asked a bold question of whether finite volume hyperbolic 3-manifolds might always admit a finite-sheeted cover which fibers over the circle. This talk will review some of the progress on this question, and discuss its relation to other questions including residual finiteness and subgroup separability, the behavior of Heegaard genus in finite-sheeted covers, CAT(0) cubings, the RFRS condition, and subgroups of right-angled Artin groups. In particular, hyperbolic 3-manifolds with LERF fundamental group are virtually fibered. Some applications of the techniques will also be mentioned.
