This work presents a shape parameterisation method based on multi-resolution subdivision curves and investigates its application to aerodynamic optimisation. Subdivision curves are defined as the limit curve of a recursive application of a subdivision rule, which provides an intrinsically hierarchical set of control polygons that can be used to provide surface control at varying levels of fidelity. In this work they are used to construct a hierarchical set of aerofoil parameterisations that can be changed throughout an optimisation procedure. This enables an optimisation to be initialised with a small number of design variables, and then periodically increased in resolution throughout. This brings the benefits of a low dimensional design space (high convergence rate, increased robustness, low cost finite-difference gradients) while still allowing the final results to be from a high-dimensional design space. In this work two approaches to subdivision aerofoil parameterisation are investigated. A multi-level refinement technique the periodically refines the parameterisation globally and an adaptive refinement scheme that refines (and coarsens) the parameterisation based on adjoint surface sensitivities. Both of these approaches are tested on a variety of optimisation problems and for each problem a range of single-level subdivision schemes (equivalent to cubic B-splines) are also used as a control group. For all the optimisation cases the multi-level and adaptive schemes converge to solutions comparable or better than the single-level methods, generally providing a significant computational advantage, and in many cases allowing a solution to be found when the single-level method would otherwise finish prematurely in a local optimum.