Tractor and Twistor fields are for conformal geometry what vector and
spinor fields are for Riemannian geometry. Tractor and Twistor spaces
and connections are usually constructed bottom-up from prolongation of
defining equations. We propose alternative top-down gauge theoretic
constructions starting from the conformal Cartan bundle and its
vectorial and spinorial associated bundles. The key ingredient is the
dressing field method of gauge symmetry reduction. This framework
leads to see Tractors and Twistors and their respective connections as
sections and gauge fields of a non-standard kind, enlightening the
deep geometric nature of these objects and that of the underlying
structure. In this seminar the case of Tractors will be fully treated,
while the case of Twistors will just be outlined at the end of the
presentation.