  We introduce the class of modules of constant Jordan type for a finite group
scheme $G$ over a field $k$ of characteristic $p > 0$. This class is closed
under taking direct sums, tensor products, duals, Heller shifts and direct
summands, and includes endotrivial modules. It contains all modules in an
Auslander-Reiten component which has at least one module in the class. Highly
non-trivial examples are constructed using cohomological techniques. We offer
conjectures suggesting that there are strong conditions on a partition to be
the Jordan type associated to a module of constant Jordan type.
