Twistor theory at fifty: from contour integrals to twistor strings

Understand

We review aspects of twistor theory, its aims and achievements spanning thelast five decades.

• In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space.
• After giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics: anti-self-duality (ASD) equations on Yang--Mills, or conformal curvature can be encoded into twistor cohomology.
• These twistor correspondences yield explicit examples of Yang--Mills, and gravitational instantons which we review.