We develop a general quantum theory for reactive collisions involving power-law potentials (-1/r^n) valid from the ultracold up to the high-temperature limit. Our quantum defect framework extends the conventional capture models to include the non-universal case when the short-range reaction probability P^{re} < 1. In the non-universal regime we study the influence of shape resonances on the collision rates. We present a method that allows to incorporate quantum corrections to the classical reaction rate due to the shape resonances and the quantum tunelling at high energies. We present explicit analytical formulas as well as numerical studies for the van der Waals (n = 6) and polarization (n = 4) potentials, showing that quantum corrections can be important even at room temperatures for typical atomic species and relate our work to recent merged beam experiments on Penning ionization for collision energies below 0.1 meV [Henson et al, Science 338, 234(2012)].