  The topological complexity TC(X) is a homotopy invariant which reflects the
complexity of the problem of constructing a motion planning algorithm in the
space X, viewed as configuration space of a mechanical system. In this paper we
complete the computation of the topological complexity of the configuration
space of n distinct points in Euclidean m-space for all m>1$ and n>1; the
answer was previously known in the cases m=2 and m odd. We also give several
useful general results concerning sharpness of upper bounds for the topological
complexity.
