These notes arose from a series of lectures first presented at the Scuola Interna- zionale Superiore di Studi Avanzati and the International Centre for Theoretical Physics in Trieste in July 1980 and then, in an extended form, at the Universities of Sofia (1980-81) and Bielefeld (1981). Their objective has been two-fold. First, to introduce theorists with some background in group representations to the notion of twistors with an emphasis on their conformal properties; a short guide to the literature on the subject is designed to compensate in part for the imcompleteness and the one-sidedness of our review. Secondly, we present a systematic study of po- sitive energy conformal orbits in terms of twistor flag manifolds. They are interpre- ted as cl assi ca 1 phase spaces of "conformal parti cl es"; a characteri sti c property of such particles is the dilation invariance of their mass spectrum which, there~ fore, consists either of the point zero or of the infinite interval 222 o < -p ~ PO - P < = The detailed table of contents should give a thorough idea of the material covered in the text. The present notes would have hardly been written without the encouragement and the support of Professor Paolo Budinich, whose enthusiasm concerning conformal semispinors (a synonym of twistors) -viewed in the spirit of El i Cartan -is having a stimulating influence.

Table of Contents:
A Guide to the List of References.- 1. The Conformal Group of a Conformally Flat Space Time and Its Twistor Representations.- 1.1 Conformal Classes of Pseudo-Riemannian Metrics.- 1.2 Connection and Curvature Forms — a Recapitulation. The Weyl Curvature Tensor.- 1.3 Global Conformal Transformations in Compactified Minkowski Space. Conformal Invariant Local Causal Order on $$ \overline {\text{M}} $$.- 1.4 The Lie Algebra of the Conformal Group and Its Twistor Representations.- 2. Twistor Flag Manifolds and SU(2,2) Orbits.- 2.1 Seven Flag Manifolds in Twistor Space. Conformal Orbits in F1 =PT.- 2.2 Points of Compactified Space-Time as 2-Planes in Twistor Space.- 2.3 An Alternative Realization of the Isomorphism $$ \overline {{\text{CM}}} \Leftrightarrow {{\text{F}}_2} $$ SU(2,2) Orbits in the Grassmann Manifold.- 2.4 Higher Flag Manifolds.- 3. Classical Phase Space of Conformal Spinning Particles.- 3.1 The Conformal Orbits F1+ and F1? as Phase Spaces of Negative and Positive Helicity O-Mass Particles.- 3.2 Canonical Symplectic Structure on Twistor Space; a Unified Phase Space Picture for Free O-Mass Particles.- 3.3 The Phase Space of Spinless Positive Mass “Conformal Particles”.- 3.4 The 10-Dimensional Phase Space of a Timelike Spinning Particle.- 3.5 The 12-Dimensional Phase Space F1,2,3?.- 4. Twistor Description of Classical Zero Mass Fields.- 4.1 Quantization of a Zero Mass Particle System: The Ladder Representations of U(2,2).- 4.2 Local Zero Mass Fields. Second Quantization.- 4.3 The Neutrino and the Photon Fields in the Twistor Picture.- 4.4 Remark on the Quantization of Higher-Dimensional Conformal Orbits.- Appendix A.Clifford Algebra Approach to Twistors. Relation to Dirac Spinors.- A.1 Clifford Algebra of O(6,?) and Bitwistor Representation of theLie Algebra SO(6,?).- A.2 The Homomorphism SL(4,?) ? SO(6,?). Inequivalent 4-Dimensional Analytic Representations of SL(4,?).- A.3 Conformal Dirac Spinors.- References.