The formalism of twistors provides a new approach to the description of basic physics. The points of Minkowski space-time are represented by 2-dimensional linear subspaces of a complex 4-dimensional vector space (flat twistor space) on which a Hermitian form of signature ++-- is defined. Free massless fields can be represented in terms of the sheaf cohomology of portions of this space. Twistor space (or a suitable part of it) can be expressed in two different ways as a complex fibration. If one or the other fibration structure is deformed, the resulting space represents not empty Minkowski space but, in one case, the general "right-flat" solution of Einstein's vacuum equations and, in the other, the general (left-handed) solution of Maxwell's equations. These provide the most primitive types of interaction (gravitational or electromagnetic) which may generalize to other fields in a comprehensive twistor scheme for the description of elementary particles.