The visible matter in the Universe is well described by the Standard Model, but this leaves open major questions in both particle physics and cosmology that may be answered by new physics at the Tera-electron-Volt range: the Terascale. The Large Hadron Collider (LHC) will soon open a new stage in humanity’s direct exploration of the fundamental physical laws at Terascale energies, which governed the evolution of the Universe a fraction of a second after the Big Bang, and are essential for understanding high-energy astrophysics. In addition to unraveling the intimate structure of matter at the Terascale, e.g., by discovering the source of particle masses and exploring matter-antimatter asymmetry, the LHC will address key cosmological issues such as how dark and conventional matter originated, which may well have been at the Terascale, and the nature of the primordial plasma that filled the Universe. This proposal will lead the understanding whatever new physics the LHC may reveal, incorporating insights from cosmology, high-energy astrophysics and speculative ideas such as string theory. This interdisciplinary approach will also facilitate the application of knowledge acquired from the LHC to fundamental cosmological and astrophysical problems, as well as illuminate future collider priorities, e.g., for LHC upgrades and/or a linear collider. This proposal will bring together particle theorists, experimentalists, astroparticle physicists and experts on field and string theory in the framework of a new ‘London Centre for Terauniverse Studies’. This will provide new opportunities for students and other young researchers to get directly involved in making LHC discoveries and exploring their implications for the Universe, and provide a mechanism for transferring to them interdisciplinary skills.