We derive new results regarding the universal asymptotic structure of spacetimes and introduce novel hyperbolic formulations of the conformal Einstein equations. These formulations permit the inclusion of generic matter fields and allow the physical spacetime dimension to be arbitrary. First, we derive coupled nonlinear wave equations by adapting the generalized harmonic gauge. Then, we deduce a first-order, symmetrizable hyperbolic equation from the corresponding reduced nonphysical Einstein equations. Additionally, we present the evolutionary form of the nonphysical constraints and discuss their solvability.