  A \textit{multicurve} $\C$ on a closed orientable surface is defined to be a
finite collection of disjoint non-isotopic essential simple closed curves. The
Dehn twist $t_{\C}$ about $\C$ is the product of the Dehn twists about the
individual curves. In this paper, we give necessary and sufficient conditions
for the existence of a root of such a Dehn twist, that is, a homeomorphism $h$
such that $h^n = t_{\C}$. We give combinatorial data that corresponds to such
roots, and use it to determine upper bounds for $n$. Finally, we classify all
such roots up to conjugacy for surfaces of genus 3 and 4.
