Summary: Discrete-velocity approximations represent a popular way for computing the Boltzmann collision operator. The direct numerical evaluation of such methods involve a prohibitive cost, typically \(O(N^{2d + 1})\) where \(d\) is the dimension of the velocity space. In this paper, following the ideas introduced in [C. Mouhot and L. Pareschi, Math. Comput. 75, No. 256, 1833–1852 (2006; Zbl 1105.76043); C. R., Math., Acad. Sci. Paris 339, No. 1, 71–76 (2004; Zbl 1054.76067)], we derive fast summation techniques for the evaluation of discrete-velocity schemes which permits to reduce the computational cost from \(O(N^{2d + 1})\) to \(O(\bar N^{d}N^{d} \log_{2}N), \bar N \ll N\), with almost no loss of accuracy.