  We investigate the effect of near threshold resonances in reactive scattering
at low energy. We find a general type of anomalous behavior of the cross
sections, and illustrate it with a real system (H$_2$ + Cl). For inelastic
processes, the anomalous energy dependence of the total cross sections is given
by $\sigma\sim\varepsilon^{-3/2}$. The standard threshold behavior given by
Wigner's law ($\sigma\sim\varepsilon^{-1/2}$) is eventually recovered at
vanishing energies, but its validity is now limited to a much narrower range of
energies. The universal anomalous behavior leads to reaction rate coefficients
behaving as $K\sim 1/T$ at low temperatures, instead of the expected constant
rate of the Wigner regime. We also provide analytical expressions for s-wave
cross sections, and discuss the implication in ultracold physics and chemistry.
