The grand challenges of contemporary fundamental physics—dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem—all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on ‘Black holes, Gravitational waves and Fundamental Physics’.

Classical gravitational bremsstrahlung in particle collisions at transplanckian energies is studied in M4 ×T d . The radiation efficiency ǫ ≡ E rad /E initial is computed in terms of the Schwarzschild radius rS( √ s), the impact parameter b and the Lorentz factor γcm and found to be ǫ = C d (rS/b) 3d+3 γ 2d+1 cm , larger than previous estimates by many powers of γcm ≫ 1. This means that in the ultrarelativistic case radiation loss becomes significant for b ≫ rS, so radiation damping must be taken into account in estimates of black hole production at transplanckian energies. The result is reliable for impact parameters in the overlap of γ ν rS < b < bc, ν = 1/2(d + 1), and b > λC , with bc marking (for d = 0) the loss of the notion of classical trajectories and λC ≡ /mc the Compton length of the scattered particles.Black hole (BH) production in LHC, predicted [1] in models with TeV-scale gravity and large extra dimensions [2-4] about ten years ago, has been the subject of intense theoretical study and numerical simulations (for a review see [5]). The prediction is based on the assumption that for impact parameters of the order of the horizon radius corresponding to the CM collision energy 2E = √ s(1) an event horizon should form due to the non-linear nature of gravity. Thewhere R is the large compactification radius. This classical, essentially, picture of BH formation is justified for transplanckian energies Indeed [6], in this case the D-dimensional Planck length l * = G D /c 3 1/(d+2) = /M * c and the de Broglie length of the collision λ B = c/ √ s satisfy the classicality condition λ B ≪ l * ≪ r S . Furthermore, gravity is believed to be the dominant force in the transplanckian region. Thus, for BH masses large compared to M * , the use of classical Einstein theory is well justified. Moreover, it seems that formation of BHs in four dimensions is predicted by string theory [7]. Thus, in spite of the fact that there are issues which require further study [8], a consensus has been reached that the prediction of BHs in ultra-high energy collisions is robust and is summarized in the widely accepted four-stage process of formation and evaporation of BHs in colliders [1,9], namely (i) formation of a closed trapped surface (CTS) in the collision of shock waves modeling the head-on particle collision, (ii) the balding phase, during which the BH emits gravitational waves and relaxes to the Myers-Perry BH, (iii) Hawking evaporation and superradiance phase in *