  Inspired by the work of Benson, Carlson, Friedlander, Pevtsova, and Suslin on
modules of constant Jordan type for finite group schemes, we introduce in this
paper the class of representations of constant Jordan type for an acyclic
quiver $Q$. We do this by first assigning to an arbitrary finite-dimensional
representation of $Q$ a sequence of coherent sheaves on moduli spaces of thin
representations. Next, we show that our quiver representations of constant
Jordan type are precisely those representations for which the corresponding
sheaves are locally free. We also construct representations of constant Jordan
type with desirable homological properties. Finally, we show that any element
of $\mathbb{Z}^L$, where $L$ is the Loewy length of the path algebra of $Q$,
can be realized as the Jordan type of a virtual representation of $Q$ of
relative constant Jordan type.
