This thesis describes the design a novel type of particle accelerator for charged particle therapy. The accelerator is called a non-scaling, Fixed Field Alternating Gradient (ns-FFAG) accelerator, and will accelerate both protons and carbon ions to energies required for clinical use. The work is undertaken as part of the PAMELA project. An existing design for a ns-FFAG is taken as a starting point and analysed in terms of its ability to suit the charged particle therapy application. It is found that this design is particularly sensitive to alignment errors and would be unable to accelerate protons and carbon ions at the proposed acceleration rate due to betatron resonance crossing phenomena. To overcome this issue, a new type of non-linear ns-FFAG is developed which avoids resonance crossing and meets the requirements provided by clinical considerations. Two accelerating rings are required, one for protons up to 250 MeV and fully stripped carbon ions to 68 MeV/u, the other to accelerate the carbon ions up to 400-430 MeV/u. Detailed studies are undertaken to show that this new type of accelerator is suitable for the application. An alignment accuracy of 50 micrometers will not have a detrimental effect on the beam and the dynamic aperture for most lattice configurations is found to be greater than 50 pi.mm.mrad normalised in both the horizontal and vertical plane. Verification of the simulation code used in the PAMELA lattice design is carried out using experimental results from EMMA, the world's first ns-FFAG for 10-20 MeV electrons built at Daresbury Laboratory, UK. Finally, it is shown that the described lattice can translate into realistic designs for the individual components of the accelerator. The integration of these components into the PAMELA facility is discussed.