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Abstract: We consider the distinguishability of multipartite quantum states by separable operations. We provide a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. Assisted with this condition, we completely solve the problem of distinguishing $(D-1)$ multipartite pure states by separable operations, where $D$ is the total dimension of the state space under consideration. Remarkably, even for the simplest composite quantum system consisting of two qubits, there exists a large class of separable operations not being realizable by local operations and classical communication. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of 1 or 2 orthogonal product states, i.e., has an orthogonal Schmidt number not less than 3.