Atmospheric retrievals of exoplanetary transmission spectra provide important constraints on various properties such as chemical abundances, cloud/haze properties, and characteristic temperatures, at the day-night atmospheric terminator. To date, most spectra have been observed for giant exoplanets due to which retrievals typically assume H-rich atmospheres. However, recent observations of mini-Neptunes/super-Earths, and the promise of upcoming facilities including JWST, call for a new generation of retrievals that can address a wide range of atmospheric compositions and related complexities. Here we report Aurora, a next-generation atmospheric retrieval framework that builds upon state-of-the-art architectures and incorporates the following key advancements: a) a generalised compositional retrieval allowing for H-rich and H-poor atmospheres, b) a generalised prescription for inhomogeneous clouds/hazes, c) multiple Bayesian inference algorithms for high-dimensional retrievals, d) modular considerations for refraction, forward scattering, and Mie-scattering, and e) noise modeling functionalities. We demonstrate Aurora on current and/or synthetic observations of hot Jupiter HD209458b, mini-Neptune K218b, and rocky exoplanet TRAPPIST1d. Using current HD209458b spectra, we demonstrate the robustness of our framework and cloud/haze prescription against assumptions of H-rich/H-poor atmospheres, improving on previous treatments. Using real and synthetic spectra of K218b, we demonstrate the agnostic approach to confidently constrain its bulk atmospheric composition and obtain precise abundance estimates. For TRAPPIST1d, 10 JWST NIRSpec transits can enable identification of the main atmospheric component for cloud-free CO$2$-rich and N$2$-rich atmospheres, and abundance constraints on trace gases including initial indications of O$3$ if present at enhanced levels ($\sim$10-100x Earth levels).