In the past 30 years a lot of work has been done on developing twistor theory.
Its creator, Roger Penrose, was first led to the concept of twistors in his investigation of the structure of spacetime and it was he who first saw the wide range of applications for this new mathematical construct.
Yet 30 years later, twistors remain relatively unknown even in the mathematical physics community.
The reason for this may be the air of mystery that seems to surround the subject even though it provides a very elegant formalism for both general relativity and quantum theory.
These notes are based on a graduate lecture course given by R. Penrose in Mathematical Institute, Oxford, in 1997 and should give a brief introduction to the basic definitions.
Let us begin with the building blocks: spinors.

Let be a vector in a (+--) spacetime.
We choose to represent it in the form of a Hermitian matrix:

Note that .
If is null future-pointing, there is a decomposition

i.e. .
Now is a spinor and is an element of the conjugate spin space.

The notation may be unusual, but all we are doing is elementary algebra.
We have shown that we can parametrise the null cone by 2 complex numbers, the components of the spinor .
In the projective case, when we are interested only in the direction of , the space of parametrisation reduces to , the 1-dimensional complex projective space, which is homeomorphic to , the celestial sphere.

Since is (up to a constant factor) the unique skew-symmetric form of maximal rank, we have the following decomposition:

where , with all diagonal entries zero.

Now we can define raising and lowering of spinor indices (nb. order of indices is important since s are skew-symmetric!):

Spinor calculations are equivalent to tensor calculations in many ways.
The main difference is that in the spinor case there are unprimed and primed (conjugate) spaces with their respective duals.
The isomorphisms between the spaces and their duals are given by the skew-symmetric form so care must be taken to write the indices in the correct order.

Note that since s are skew.