During the past two decades, Roger Penrose's twistor theory has been a continuing source of mathematical inspiration. This ambitious theory aims to reformulate the foundations of physics using conformal and complex geometry. At the heart of twistor theory lies a geometrical transform know as the Penrose transform. This book provides coverage of this transform in a general setting, utilizing complex homogeneous spaces, with mathematical input taken from the representation theory of Lie groups. The Penrose Transform gives readers much original research that is not available in any other text, with special attention to the twistor theory of Minkowski space.