  We provide algorithms to count and enumerate representatives of the (right)
ideal classes of an Eichler order in a quaternion algebra defined over a number
field. We analyze the run time of these algorithms and consider several related
problems, including the computation of two-sided ideal classes, isomorphism
classes of orders, connecting ideals for orders, and ideal principalization. We
conclude by giving the complete list of definite Eichler orders with class
number at most 2.
