Heisenberg quantum spin chains and interacting Bose gases in one dimension are two fundamental exactly solvable models of strongly-correlated physics with direct experimental relevance. Recent progresses in the theory of integrable models have opened the door to extremely accurate computations of their dynamical correlation functions, which are experimentally accessible via either inelastic neutron scattering (for spin systems) or Bragg spectroscopy (for quantum gases in optical lattices). We review these developments and their application in the description of experimental data on representative systems.