We review the first modern singularity theorem, published by Penrose in 1965. This is the first genuine post-Einsteinian result in general relativity, where the fundamental and fruitful concept of the closed trapped surface was introduced. We include historical remarks, an appraisal of the theorem's impact, and relevant current and future work that belongs to its legacy.

A purely algebraic construction of super-energy tensors for arbitrary fields is presented in any dimensions. These tensors have good mathematical and physical properties, and they can be used in any theory having as basic arena an n-dimensional manifold with a metric of Lorentzian signature. In general, the completely timelike component of these s-e tensors has the mathematical features of an energy density: they are positive definite and satisfy the dominant property. Similarly, the super-momentum vectors have mathematical properties of s-e flux vectors. The classical Bel-Robinson tensor is included in our general definition. The energy-momentum and super-energy tensors of physical fields are also obtained, and the procedure is illustrated by writing down these tensors explicitly for the cases of scalar, electromagnetic, and Proca fields. Moreover, `(super)$^k$-energy' tensors are defined and shown to be meaningful and in agreement for the different physical fields. In flat spacetimes, they provide infinitely many conserved quantities. In non-flat spacetimes conserved s-e currents are found for any minimally coupled scalar field whenever there is a Killing vector. Furthermore, the exchange of gravitational and electromagnetic super-energy is also shown by studying the propagation of discontinuities.

I present the junction conditions for $F(R)$ theories of gravity and their implications: the generalized Israel conditions and equations. These junction conditions are necessary to construct global models of stars, galaxies, etc., where a vacuum region surrounds a finite body in equilibrium, as well as to describe shells of matter and braneworlds, and they are stricter than in General Relativity in both cases. For the latter case, I obtain the field equations for the energy-momentum tensor on the shell/brane, and they turn out to be, remarkably, the same as in General Relativity. An exceptional case for quadratic $F(R)$, previously overlooked in the literature, is shown to arise, allowing for a discontinuous $R$ and leading to an energy-momentum content on the shell with unexpected properties, such as nonvanishing components normal to the shell and a new term resembling classical dipole distributions. For the former case, they require not only the agreement of the first and second fundamental forms on both sides of the matching hypersurface but also that the scalar curvature $R$ and its first derivative $\nabla R$ agree there too. I argue that, as a consequence, matched solutions in General Relativity are not solutions of $F(R)$ models generically. Several relevant examples are analyzed.

A new class of exact solutions to Einstein’s field equations with a perfect-fluid source is presented. The solutions describe spatially inhomogeneous cosmological models and have a realistic equation of state p=ρ/3. The properties of the solutions are discussed. The most remarkable feature is the absence of an initial singularity, the curvature and matter invariants being regular and smooth everywhere. We also present an alternative interpretation of the solution as a globally regular cylindrically symmetric space-time.