We obtain the weak Harnack inequality and Holder estimates for a large class of kinetic integro-differential equations. We prove that the Boltzmann equation without cutoff can be written in this form and satisfies our assumptions provided that the mass density is bounded away from vacuum and mass, energy and entropy densities are bounded above. As a consequence, we derive a local Holder estimate and a quantitative lower bound for solutions of the (inhomogeneous) Boltzmann equation without cutoff .