Let $M$ be a compact, connected surface without boundary and different from $\rp$, and let $B_n(M)$ and $P_n(M)$ denote its braid group and pure braid group on $n$ strings respectively. In this paper, we study the roots of the ‘full twist’ braid in $P_n(M)$ and $B_n(M){\setminus} P_n(M)$. Our main results may be summarised as follows: first, the full twist has no non-trivial root in $P_n(M)$. Further, if $M\ne \St$ and $k\geq 2$, it has a $k\th$ root in $B_n(M){\setminus} P_n(M)$ if and only if $k$ divides either $n$ or $n-1$. This generalises results concerning the sphere of Gillette, Van Buskirk and Murasugi. We also show that the Artin pure braid groups and the pure braid groups of the sphere admit a (non-trivial) splitting as a direct product of which one of the factors is the cyclic group generated by the full twist.