This monograh is mainly devoted to the theory of integral representation in convex cones due to G. Choquet and to several of its applications to Analysis: Classical theorems of Bochner-Weil and of Berstein, theorem of Choquet-Deny, axiomatics of Brelot an Bauer in potential theory, results of Talagrand concerning invariant measures and capacities and those of Royer and Yor concerning Quasi-invariant measures in field theory. This book is accessible to a student with a Master's degreesM1 and M2, with emphasis on analyses, and can be considered as an introduction to the richness and variety of the subjects. The three tools created by G. Choquet in this framework (weakly complete cones, caps, conical measures) are also studied for themselves. Moreover, there are also useful in other fields close to Analysis : zonoforms and vector measures, statistical decision spaces bireticulated cones and normal cones contained in Banach spaces. Remark that in the postface G. Choquet relates the thought process which led him to find out the three tools mentioned above.