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High-fidelity planar quantum circuits have been recently experimentally demonstrated, enabling a new generation of quantum computing, based on reliable, monolithically integrated planar optical devices. The present paper is a contribution to the development of this new concept, giving a general and at most comprehensive overview of quantum algorithms that can be implemented using planar optics devices. Starting from the four postulates of quantum mechanics, the complete set of quantum-gate operations is described using only planar 1-qubit rotations and CNOT gates, both in universal and non-deterministic forms. Both single-photon and coherent-states quantum computing are considered, and the effects of photon losses and phase noise are investigated.