  This paper is a review of the twistor theory of irreducible G-structures and
affine connections. Long ago, Berger presented a very restricted list of
possible irreducibly acting holonomies of torsion-free affine connections. His
list was complete in the part of metric connections, while the situation with
holonomies of non-metric torsion-free affine connections was and remains rather
unclear. One of the results discussed in this review asserts that any
torsion-free holomorphic affine connection with irreducibly acting holonomy
group can, in principle, be constructed by twistor methods. Another result
reveals a new natural subclass of affine connections with "very little torsion"
which shares with the class of torsion-free affine connections two basic
properties --- the list of irreducibly acting holonomy groups of affine
connections in this subclass is very restricted and the links with the twistor
theory are again very strong.
