We evaluate the dynamic structure factor $S(q,omega)$ 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 $S(q,omega)$. The sharp peak $Spropto qdelta(omega-uq)$, characteristic for the Tomonaga-Luttinger model, broadens up; $S(q,omega)$ for a fixed $q$ becomes finite at arbitrarily large $omega$. The main spectral weight, however, is confined to a narrow frequency interval of the width $deltaomegasim q^2/m$. 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 $q$.
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