  We determine a set of generators for the Brunnian braids on a general surface
$M$ for $M\not=S^2$ or $\RP^2$. For the case $M=S^2$ 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.
