We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then `quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP^3, compactified Minkowski space CMh and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on CMh pulls back to the basic instanton on S^4\subset CMh and that this observation quantises to obtain the Connes-Landi instanton on ��-deformed S^4 as the pull-back of the tautological bundle on our ��-deformed CMh. We likewise quantise the fibration CP^3--> S^4 and use it to construct the bundle on ��-deformed CP^3 that maps over under the transform to the ��-deformed instanton.

68 pages 0 figures. Significant revision now has detailed formulae for classical and quantum CP^3