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This survey paper presents lecture notes from a series of four lectures addressed to a wide audience and it offers an introduction to several topics in conformal differential geometry. In particular, a very nice and gentle introduction to the conformal Riemannian structures themselves, flat or curved, is presented in the beginning. Then the behavior of the covariant derivatives under the rescaling of the metrics is described. This leads to Penrose's local twistor transport which is introduced in the approach credited to T. Y. Thomas. The resulting object is called the tractor connection and is equivalent to Cartan's concept of connections on generalized spaces [see also {\it T. N. Bailey, M. G. Eastwood} and {\it R. A. Gover}, Rocky Mt. J. Math. 24, 1191-1217 (1994; Zbl 0828.53012)]. \par The third lecture builds explicit links to the representation theory and describes the invariant operators on the homogeneous vector bundles over the round sphere (the homogeneou!
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