The purpose of this paper is to investigate the extension of the Luenberger observer design approach to linear quantum stochastic systems to obtain coherent quantum observers. We show how a physically realizable quantum observer can be designed, consistent with the laws of quantum mechanics. The quantum observer has the property that the mean values of the observer variables asymptotically track the corresponding mean values of the plant. In addition, we discuss entanglement of the joint plant-observer state, as well as the implications of the no-cloning theorem. Several examples are considered.