We compute the spectral function rho(q,omega) of the one- dimensional Luttinger model. We discuss the distinct influences of charge-spin separation and of the anomalous dimensions of the fermion operators and their evolution with correlation strength. Charge-spin separation shows up in finite spectral weight at frequencies between v_sigma * q and v_rho * q where v_rho and v_sigma are the velocities of charge and spin fluctuations while spectral weight above v_rho * q and below -v_rho * q is generated by the hybridization of the Fermi surface at +/-k_F by interactions. There are nonuniversal power-law singularities at these special frequencies. We discuss the consistency of recent photoemission experiments on low-dimensional conductors with a Luttinger liquid picture which then would suggest very strong long- range interactions. It is pointed out that many-particle correlation functions in principle exhibit similar singularities but they probe different and complementary aspects of the Fermi surface interactions.