Having established the basic notions of manifolds and fibre bundles associated with vector spaces, this chapter is an essential prerequisite for many geometrical definitions introduced in subsequent chapters. These will often be used to reformulate the theories of Maxwellian electrodynamics (of fields and particles) and (generalised) Einsteinian gravitation in tensorial form. They will then enable one to establish a relativistic framework for classical observers and their neighbours in a general spacetime context.