This thesis aims at shining some new light on the terra incognita of multi-dimensional hyperbolic systems of conservation laws by means of techniques whose application to this ﬁeld is a brand new idea. In particular, our attention focuses on the isentropic compressible Euler equations of gas dynamics, the oldest but yet most prominent para- digm for this class of equations. The theory of the Cauchy problem for hyperbolic systems of conservation laws in more than one space dimension is still in its dawning and has been facing some basic is- sues so far: do there exist weak solutions for any initial data? how to prove well-posedness for weak solutions? which is a good space for a well-posedness theory? are entropy inequalities good selection criteria for uniqueness? Inspired by these interesting questions, we obtained some new results here collected. First, we present a counterexample to the well-posedness of entropy solutions to the Cauchy problem for the multi-dimensional compressible Euler equations: in our construc- tion the entropy condition is not suﬃcient as a selection criterion for unique solutions. Furthermore, we show that such a non-uniqueness theorem holds also for some Lipschitz initial data in two space dimen- sions. Our results and constructions build upon the method of convex integration developed by De Lellis-Sz´kelyhidi for the incompressible e Euler equations and based on a revisited “h-principle”. Finally, we prove existence of weak solutions to the Cauchy problem for the isentropic compressible Euler equations in the particular case of regular initial density.