  Let Mod_{g,b} denote the mapping class group of a surface of genus g with b
punctures. Feng Luo asked in a recent preprint if there is a universal upper
bound, independent of genus, for the number of torsion elements needed to
generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements
suffice to generate Mod_{g,0}. We also prove the more delicate result that
there is an upper bound, independent of genus, not only for the number of
torsion elements needed to generate Mod_{g,b} but also for the order of those
elements. In particular, our main result is that 6 involutions (i.e.
orientation-preserving diffeomorphisms of order two) suffice to generate
Mod_{g,b} for every genus g >= 3, b = 0, and g >= 4, b = 1.
