This technical note describes an application of nonlinear controllability theory to the problem of spacecraft attitude control using control moment gyroscopes (CMGs). Nonlinear controllability theory is used to show that a spacecraft carrying one or more CMGs Is controllable on every angular momentum level set In spite of the presence of singular CMG configurations, that Is, given any two states having the same angular momentum, any one of them can be reached from the other using suitably chosen motions of the CMG gimbals. This result is used to obtain sufficient conditions on the momentum volume of the CMG array that guarantee the existence of gimbal motions which steer the spacecraft to a desired spin state or rest attitude.