  This paper investigates the tilting modules of the cyclotomic q-Schur
algebras, 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 certain permutation modules. Surprisingly, Weyl filtrations --
which are in general more powerful than Specht filtrations -- play only a
secondary role.
