This paper investigates the tilting modules of the cycloto mic q-Schur alge- bras, the Young modules of the Ariki-Koike algebras, and the interconnections between them. The main tools used to understand the tilting modules are contragredient duality, and the Specht filtrations and dual Specht filtrations of cert ain permutation modules. Sur- prisingly, Weyl filtrations — which are in general more power ful than Specht filtrations — play only a secondary role.