Affine Connections on Involutive G-structures

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.