This manuscrit is a project of book on Hamilton-Jacobi Equations and Control Problems with discontinuities. For the time being, it is composed of 4 parts: -- Part I is a toolbox with key results which are used in all the other parts. -- Part II describes several approaches which were introduced recently to treat problems involving co-dimension 1 discontinuities or networks. -- Part III concerns stratified problems in $R^N$, i.e. problems with discontinuities of any co-dimensions. -- Part IV addresses the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. This first version is undoubtedly still imperfect and we will welcome any (fair) comment on it: while, to our point of view, Part I and II are in a rather satisfactory form, Part III can certainly be improved and Part IV is still in construction: we certainly wish to incorporate more applications. We are ready to accept contributions (provided they fit with the scope and style of the book), in particular those concerning applications. Of course, we will give credit, one way or the other, to these contributions.