  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. They also underlie the twistor
approach to integrability: the solitonic systems arise as symmetry reductions
of ASD Yang--Mills equations, and Einstein--Weyl dispersionless systems are
reductions of ASD conformal equations.
  We then review the holomorphic string theories in twistor and ambitwistor
spaces, and explain how these theories give rise to remarkable new formulae for
the computation of quantum scattering amplitudes. Finally we discuss the
Newtonian limit of twistor theory, and its possible role in Penrose's proposal
for a role of gravity in quantum collapse of a wave function.
