A first approximation to the idea of a foliation is a dynamical system,
and the resulting decomposition of a domain by its trajectories. This is
an idea that dates back to the beginning of the theory of differential
equations, i.e. the seventeenth century. Towards the end of the
nineteenth century, Poincare developed methods for the study of global,
qualitative properties of solutions of dynamical systems in situations
where explicit solution methods had failed: He discovered that the study
of the geometry of the space of trajectories of a dynamical system
reveals complex phenomena. He emphasized the qualitative nature of these
phenomena, thereby giving strong impetus to topological methods. A
second approximation is the idea of a foliation as a decomposition of a
manifold into submanifolds, all being of the same dimension. Here the
presence of singular submanifolds, corresponding to the singularities in
the case of a dynamical system, is excluded. This is the case we treat
in this text, but it is by no means a comprehensive analysis. On the
contrary, many situations in mathematical physics most definitely
require singular foliations for a proper modeling. The global study of
foliations in the spirit of Poincare was begun only in the 1940's, by
Ehresmann and Reeb.