Using the thin-shell formalism we discuss the motion of domain walls in de
Sitter and anti-de Sitter (AdS) time-dependent bulks. This motion results i
n a dynamics for the brane scale factor. We show that in the case of a clea
n brane the scale factor describes both singular and nonsingular universes,
with phases of contraction and expansion. These phases can be understood a
s quotients of AdS spacetime by a discrete symmetry group. We discuss this
effect in some detail, and suggest how the AdS/CFT correspondence could be
applied to obtain a nonperturbative description of brane-world string cosmo
logy.