Let Mod(Sigma(g,p)) be the mapping class group of a closed oriented surface Sigma(g,p) of genus g >= 1 with p punctures. Wajnryb proved that Mod(Sigma(g,0)) is generated by two elements. Korkmaz proved that one of these generators may be taken to be a Dehn twist. Korkmaz also proved the same result in the case of Mod(Sigma(g,1)). For p >= 2, we prove that Mod(Sigma(g,p)) is generated by three elements.