Distributions, i.e., subsets of tangent bundles formed by piecing together
subspaces of tangent spaces, are commonly encountered in the theory and
application of differential geometry. Indeed, the theory of distributions is a
fundamental part of mechanics and control theory.
The theory of distributions is presented in a systematic way, and
self-contained proofs are given of some of the major results. Parts of the
theory are presented in the context of generalised subbundles of vector
bundles. Special emphasis is placed on understanding the r\^ole of sheaves and
understanding the distinctions between the smooth or finitely differentiable
cases and the real analytic case. The Orbit Theorem and applications, including
Frobenius's Theorem and theorems on the equivalence of families of vector
fields, are considered in detail. Examples illustrate the phenomenon that can
occur with generalised subbundles and distributions