The topological complexity was introduced by Michael Farber in 2003 motivated by applications of algebraic topology to robotics. It is a numerical homotopy invariant of a space which measures the instability of motion planning. In this thesis we determine this invariant for unordered configuration spaces of surfaces in many cases and reduce it to a few possible values in other cases. We also determine the topological complexity of mixed configuration spaces and related spaces. In contrast to the ordered configuration spaces, these computations remained elusive because the standard methods do not work here, as we argue in the Appendix. Apart from the interest from the motion planning perspective to decide whether unordered or ordered configurations have a higher topological complexity, there is another motivation. Namely, it is an open problem to give an algebraic description of the topological complexity of an aspherical space in terms of the fundamental group. The spaces under consideration are aspherical and so the topological complexity (being a homotopy invariant) becomes an invariant of their fundamental groups, the surface braid groups. The computation of the topological complexity of surface braid groups and their finite index subgroups thus provides further examples which might help tackle this open problem. Furthermore, the results could be used to gain information about the subgroup structure of surface braid groups. Often the topological complexity is calculated indirectly without actually finding an optimal motion planner which realizes it. Nonetheless, in some cases we will construct explicit motion planners and then prove that they are optimal. All those motion planners are collected in the last chapter.