Geometric quantization allows one to turn symplectic manifolds into Hilbert spaces. In physical systems it allows one to pass from the phase space of classical mechanics to the space of quantum states. Higher-dimensional quantum field theories are described by a higher-categorical information while higher-dimensional classical field theories are described by ``higher'' symplectic structures; in this talk these are shifted symplectic structures. I will explain how to apply geometric quantization to shifted symplectic structures to extract such a categorical information. I will give several examples of this procedure including Hamiltonian G-spaces and moduli spaces of flat connections.