This third version is a major release of our book project on Hamilton-Jacobi Equations and Control Problems with discontinuities. Compared to the second version (online in december 2019), we have completely reorganized the manuscript, added new results and examples, changed some points-of-view, detailed some proofs and corrected several mistakes. Version 2 had 353 pages, this one 550. Most probably this is the last version before the final one. It is now composed of six parts: Part I is still a toolbox with key results which are used in all the other parts. The study of the simplest case, i.e. the case of a co-dimension 1 discontinuity, is now split in two parts: in Part II, we only consider control problems and the associated Bellman Equations are treated by using only the classical notion of viscosity solutions. In this part, the methods are a combinations of control and pdes techniques. On the contrary, Part III describes purely pdes approaches which are inspired by the literature on Hamilton-Jacobi Equations on networks and which can handle the case of non-convex Hamiltonians. In this part, we present two notions of solutions, namely flux-limited and junction viscosity solutions, and we study in detail their properties by providing comparison and stability results. We also show that they are ``almost'' equivalent when both make sense, i.e. for quasi-convex Hamiltonians. Part IV concerns stratified problems in $\R^N$, i.e. problems with discontinuities of any co-dimensions: the main change compared to the previous version is the introduction of a notion of ``weak'' stratified (sub)solution. In Part V, we address the case of stratified problems in bounded or unbounded domains with state-constraints, allowing very surprising applications as well as singular boundary conditions. Finally, in Part VI we describe some applications to KPP (Kolmogorov-Petrovsky-Piskunov) type problems and we discuss possible extensions to problems with jumps and to ``stratified networks''. We consider this third version as being finalized until Part IV, while Part V and VI may need some further improvements. We hope to have a completely final version by june 2023. All comments are welcome!