After briefly discussing the non-impulsive (smooth) case, we will show that every impulsive wave-type spacetime of the form $M=N\times R^2_1$, with line element $ds^2 = dh^2 + 2 du dv + f(x)\delta(u) du^2$ is geodesically complete. Here $(N,h)$ is an arbitrary connected, complete Riemannian manifold, $f$ is a smooth function and $\delta$ denotes the Dirac distribution on the hypersurface $u=0$. Moreover the geodesics behave as is physically expected.
In this talk I will present the idea of a geometrical interpretation of dark phenomena on cosmological and astrophysical scales. In particular, I will review the status of our knowledge on the physics of extensions of General Relativity in relation with dark matter and dark energy and I will delineate the new challenges in this research field for the next years.