Extremal optimisation is an emerging nature inspired meta-heuristic search technique that allows a poorly performing solution component to be removed at each iteration of the algorithm and replaced by a random value. This creates opportunity for assignment type problems as it enables a component to be moved to a more appropriate group. This may then drive the system towards an optimal solution. In this chapter, the general capabilities of extremal optimisation, in terms of assignment type problems, are explored. In particular, we provide an analysis of the moves selected by extremal optimisation and show that it does not suffer from premature convergence. Following this we develop a model of extremal optimisation that includes techniques to a) process constraints by allowing the search to move between feasible and infeasible space, b) provide a generic partial feasibility restoration heuristic to drive the solution towards feasible space, and c) develop a population model of the meta-heuristic that adaptively removes and introduces new members in accordance with the principles of self-organised criticality. A range of computational experiments on prototypical assignment problems, namely generalised assignment, bin packing, and capacitated hub location, indicate that extremal optimisation can form the foundation for a powerful and competitive meta-heuristic for this class of problems.