This work is an expanded version of lectures given at Ecole Normale Supérieure (France) and Universidade Federal de Sgo Carlos (Brasil) with the following two aims :
- Introduce some basic tools of differential geometry (Ehresmann's jets, Whitney's extension theorem, Thom's transversality theorem, Moser's infinitesimal method, contact and symolectic structures) in a fashion suitable for applications.

- Use these tools in the study of singularities of smooth actions (normal forms, various versions of Sternberg's theorem, linearisation of abelian group actions, singular integral manifolds of p.d.e'.s, etc...).
One of our goals was to show that rather deep results in the local theory of actions could be obtained using quite simple geometric ideas ans very little analysis.
In addition to the examples included in the bulk of the text, a substantial conclusion and several appendices should enable the reader to appreciate the impli-cations of our results and the richness of the subject as a whole.