This work presents the aerodynamic topology optimisation of high pressure turbine rotor blade tips. Before carrying out the topology optimisation on the blade tip, some initial tip design studies were carried out. The winglet shape was optimised using two different design space setups and parameter limits. The optimum winglet design features the largest overhangs and in the case of unconstrained optimisation proved to have 1.40% greater aerodynamic efficiency. Secondly, a radial basis function based parametrisation was set up to allow the creation of single squealer line using the flat tip blade as a baseline geometry. The optimum case proved to increase efficiency 0.46% compared to the flat tip. After that, a combination of winglet and topology free squealer tips was investigated for topology optimisation. The winglet tip was parametrized as in the winglet only optimisation cases and topology free squealer walls were created using mapping of radial basis function surfaces of different complexities. It is shown that by combining both winglet and novel squealer topology optimisation, better designs of different topologies can be produced.