Let K be a field with a derivation D and let K[y,D] denote the ring of differential polynomials in the indeterminate y with coefficients in K. The object of this paper is to study the torsion modules over the class of differential polynomial rings. It is known that if K is a universal field, then the torsion modules are all semisimple, injective, and there is exactly one isomorphism class of simple modules. In this paper it is shown that this situation is a rather uncommon one. For most examples of K the torsion indecomposable modules over K[y,D] have a very complicated structure. For example, it is shown that if K is any subfield of the field of meromorphic functions over the complex numbers containing the rationals, there exist torsion indecomposable injective, cyclic modules whose lattice of submodules is not linearly ordered. Generally speaking, it is shown that rings of differential polynomials over fields which have lots of solutions to specified systems of differential equations will have a simpler structure for their indecomposable torsion modules than those polynomial rings over fields which contain few solutions.