We provide a new general scheme for the geometric quantisation of Sp(1)-symmetric hyper-Kähler manifolds, considering Hilbert spaces of holomorphic sections with respect to the complex structures in the hyper-Kähler 2-sphere. Under properness of an associated moment map, or other finiteness assumptions, we construct unitary quantum (super) representations of central extensions of certain subgroups of Riemannian isometries preserving the 2-sphere, and we study their decomposition in irreducible components. We apply this quantisation scheme to hyper-Kähler vector spaces, the Taub-NUT metric on R 4 , moduli spaces of framed SU(r)-instantons on R 4 , and partly to the Atiyah-Hitchin manifold of magnetic monopoles in R 3 Contents 1. Introduction 1 Acknowledgements 5 2. Abstract Sp(1)-symmetric hyper-Kähler quantisation 6 3. Examples of applications 15 4. Outlook and further perspectives 25 Appendix A. Comparison with the standard approach 26 References 26