The main topic of these notes is the theory of connections. There are two basic notions in the theory: the notion of covariant derivation which concerns differentiable sections of vector bundles, and the notion of connection forms on principal bundles. These two notions are by no means independent of each other. While any law of соvariant derivation in a vector bundle can be defined by a connection form in the principal bundle of frames, an independent treatment of crvariart derivations is desirable in view of many applications whore the principal bundle remains in the background.