A1 algebraic topology over a field This text deals with A¹ homotopy theory over a base field i.e. with the natural homotopy theory associated to the category of smooth varieties over a field in which the affine line is imposed to be contractible. It is a natural sequel to the foundational paper on A¹ homotopy theory written together with V. Voevodsky. Inspired by classical results in algebraic topology we present new techniques new results and applications related to the properties and computations of A¹ homotopy sheaves A¹ homotogy sheaves and sheaves with generalized transfers as well as to algebraic vector bundles over affine smooth varieties