  The spin foam formalism provides transition amplitudes for loop quantum
gravity. Important aspects of the dynamics are understood, but many open
questions are pressing on. In this paper we address some of them using a
twistorial description, which brings new light on both classical and quantum
aspects of the theory. At the classical level, we clarify the covariant
properties of the discrete geometries involved, and the role of the simplicity
constraints in leading to SU(2) Ashtekar-Barbero variables. We identify areas
and Lorentzian dihedral angles in twistor space, and show that they form a
canonical pair. The primary simplicity constraints are solved by simple
twistors, parametrized by SU(2) spinors and the dihedral angles. We construct
an SU(2) holonomy and prove it to correspond to the (lattice version of the)
Ashtekar-Barbero connection. We argue that the role of secondary constraints is
to provide a non trivial embedding of the cotangent bundle of SU(2) in the
space of simple twistors. At the quantum level, a Schroedinger representation
leads to a spinorial version of simple projected spin networks, where the
argument of the wave functions is a spinor instead of a group element. We
rewrite the Liouville measure on the cotangent bundle of SL(2,C) as an integral
in twistor space. Using these tools, we show that the
Engle-Pereira-Rovelli-Livine transition amplitudes can be derived from a path
integral in twistor space. We construct a curvature tensor, show that it
carries torsion off-shell, and that its Riemann part is of Petrov type D.
Finally, we make contact between the semiclassical asymptotic behaviour of the
model and our construction, clarifying the relation of the Regge geometries
with the original phase space.
