  Let B_n(RP^2)$ (respectively P_n(RP^2)) denote the braid group (respectively
pure braid group) on n strings of the real projective plane RP^2. In this paper
we study these braid groups, in particular the associated pure braid group
short exact sequence of Fadell and Neuwirth, their torsion elements and the
roots of the `full twist' braid. Our main results may be summarised as follows:
first, the pure braid group short exact sequence
  1 --> P_{m-n}(RP^2 - {x_1,...,x_n}) --> P_m(RP^2) --> P_n(RP^2) --> 1
  does not split if m > 3 and n=2,3. Now let n > 1. Then in B_n(RP^2), there is
a k-torsion element if and only if k divides either 4n or 4(n-1). Finally, the
full twist braid has a k-th root if and only if k divides either 2n or 2(n-1).
