Most celestial bodies, from planets, to stars, to black holes; gain mass during their lives by means of an accretion disc. Understanding the physical processes that determine the rate at which matter accretes and energy is radiated in these discs is vital for unraveling the formation, evolution, and fate of almost every type of object in the Universe. Despite the fact that magnetic fields have been known to be crucial in accretion discs since the early 90’s, the majority of astrophysical questions that depend on the details of how disc accretion proceeds are still being addressed using the “standard” accretion disc model (developed in the early 70’s), where magnetic fields do not play an explicit role. This has prevented us from fully exploring the astrophysical consequences and observational signatures of realistic accretion disc models, leading to a profound disconnect between observations (usually interpreted with the standard paradigm) and modern accretion disc theory and numerical simulations (where magnetic turbulence is crucial). The goal of this proposal is to use several complementary approaches in order to finally move beyond the standard paradigm. This program has two main objectives: 1) Develop the theoretical framework to incorporate magnetic fields, and the ensuing turbulence, into self-consistent accretion disc models, and investigate their observational implications. 2) Investigate transport and radiative processes in collision-less disc regions, where non-thermal radiation originates, by employing a kinetic particle description of the plasma. In order to achieve these goals, we will use, and build upon, state-of-the-art magnetohydrodynamic and particle-in-cell codes in conjunction with theoretical modeling. This framework will make it possible to address fundamental questions on stellar and planet formation, binary systems with a compact object, and supermassive black hole feedback in a way that has no counterpart within the standard paradigm.