This two-volume monograph obtains fundamental notions and results of the standard differential geometry of smooth (C^INFINITY) manifolds, without using differential calculus. Here, the sheaf-theoretic character is emphasised. This has theoretical advantages such as greater perspective, clarity and unification, but also practical benefits ranging from elementary particle physics, via gauge theories and theoretical cosmology (differential spaces'), to non-linear PDEs (generalised functions). Thus, more general applications, which are no longer smooth' in the classical sense, can be...