Scalable quantum information processing will require quantum networks of qubits with the ability to coherently transfer quantum states between the desired sender and receiver nodes. Here we propose a scheme to implement a quantum router that can direct quantum states from an input qubit to a preselected output qubit. The path taken by the transferred quantum state is controlled by the state of one or more ancilla qubits. This enables both directed transport between a sender and a number of receiver nodes, and generation of distributed entanglement in the network. We demonstrate the general idea using a two-output setup and discuss how the quantum routing may be expanded to several outputs. We also present a possible realization of our ideas with superconducting circuits.