  We define a "quantum relation" on a von Neumann algebra M \subset B(H) to be
a weak* closed operator bimodule over its commutant M'. Although this
definition is framed in terms of a particular representation of M, it is
effectively representation independent. Quantum relations on l^\infty(X)
exactly correspond to subsets of X^2, i.e., relations on X. There is also a
good definition of a "measurable relation" on a measure space, to which quantum
relations partially reduce in the general abelian case. By analogy with the
classical setting, we can identify structures such as quantum equivalence
relations, quantum partial orders, and quantum graphs, and we can generalize
Arveson's fundamental work on weak* closed operator algebras containing a masa
to these cases. We are also able to intrinsically characterize the quantum
relations on M in terms of families of projections in M \otimes B(l^2).
