We determine a set of generators for the Brunnian braids on a general surface M for M 6 D S 2 or RP 2 . For the case M D S 2 or RP 2 , a set of generators for the Brunnian braids on M is given by our generating set together with the homotopy groups of a 2-sphere.57M07, 57M99; 20F36, 55Q40