We evaluate the dynamic structure factor S(q,ω) of interacting one-dimensional spinless fermions with a nonlinear dispersion relation. The combined effect of the nonlinear dispersion and of the interactions leads to new universal features of S(q,ω). The sharp peak S(q,ω) αqδ(ω-uq), characteristic for the Tomonaga-Luttinger model, broadens up; S(q,ω) for a fixed q becomes finite at arbitrarily large ω. The main spectral weight, however, is confined to a narrow frequency interval of the width δω∼q2/m. At the boundaries of this interval the structure factor exhibits power-law singularities with exponents depending on the interaction strength and on the wave number q.