  We determine the topological complexity of unordered configuration spaces on
almost all punctured surfaces (both orientable and non-orientable). We also
give improved bounds for the topological complexity of unordered configuration
spaces on all aspherical closed surfaces, reducing it to three possible values.
The main methods used in the proofs were developed in 2015 by Grant, Lupton and
Oprea to give bounds for the topological complexity of aspherical spaces. As
such this paper is also part of the current effort to study the topological
complexity of aspherical spaces and it presents many further examples where
these methods strongly improve upon the lower bounds given by zero-divisor
cup-length.
