We consider the Cauchy problem for the multi-dimensional compressible Euler equations, evolving from an open set of compressive and generic smooth initial data. We construct unique solutions to the Euler equations which are as smooth as the initial data, in the maximal spacetime set characterized by: at any point in this spacetime, the solution can be smoothly and uniquely computed by tracing both the fast and slow acoustic characteristic surfaces backward-in-time until reaching the Cauchy data prescribed along the initial time-slice. This spacetime is sometimes referred to as the "maximal globally hyperbolic development" (MGHD) of the given Cauchy data. We prove that the future temporal boundary of this spacetime region is a singular hypersurface, consisting of the union of three sets: first, a co-dimension-2 surface of "first singularities" called the pre-shock; second, a downstream co-dimension-1 surface emanating from the pre-shock, on which the Euler solution experiences a continuum of gradient catastrophes; third, an upstream co-dimension-1 surface consisting of a Cauchy horizon emanating from the pre-shock, which the Euler solution cannot reach. In order to establish this result, we develop a new geometric framework for the description of the acoustic characteristic surfaces, and combine this with a new type of differentiated Riemann-type variables which are linear combinations of gradients of velocity/sound speed and the curvature of the fast acoustic characteristic surfaces. This is a joint work with Steve Shkoller (UC Davis).