We review the theoretical foundations, methodological approaches and current status of the determination of nuclear parton distribution functions (PDFs). A large variety of measurements in fixed-target and collider experiments provide increasingly precise constraints on various aspects of nuclear PDFs, including shadowing, antishadowing, the EMC effect, Fermi motion, flavour separation, deuteron binding, target-mass and other higher twist effects. We give particular emphasis to measurements carried out in proton-lead collisions at the Large Hadron Collider, which have revolutionised the global analysis during the past decade. These measurements include electroweak boson, isolated photon, single inclusive hadron, jet, and heavy-flavour observables.