  The cycles of a graph give a natural cyclic ordering to their edge-sets, and
these orderings are consistent in that two edges are adjacent in one cycle if
and only if they are adjacent in every cycle in which they appear together. An
orderable matroid is one whose set of circuits admits such a consistent
ordering. In this paper, we consider the question of determining which matroids
are orderable. Although we are able to answer this question for non-binary
matroids, it remains open for binary matroids. We give examples to provide
insight into the potential difficulty of this question in general. We also show
that, by requiring that the ordering preserves the three arcs in every
theta-graph restriction of a binary matroid $M$, we guarantee that $M$ is
orderable if and only if $M$ is graphic.
