The field of exoplanetary sciences has grown from an era of detection to one of characterization. To date, over 4000 exoplanets have been discovered and over 50 of them have been observed with primary transit spectroscopy methods. The current population of characterized exoplanets spans a wide range of parameter space; from ultra-hot Jupiters with atmospheric temperatures beyond 3000 K, to temperate mini Neptunes that may host water in their atmospheres. Upcoming observational facilities in the next two decades will deliver exquisite spectra of exoplanet atmospheres at wavelengths never probed before, with unprecedented precision, and at much higher resolution than currently possible, effectively expanding the number of exoplanets with observed spectra. Nonetheless, an increasingly diverse planet population and higher fidelity data necessarily demand more flexible, complex, and generalized modeling frameworks. In this thesis, we present our work on atmospheric retrievals of exoplanets, focusing on investigating the robustness of the model assumptions inevitably employed to infer basic planetary conditions, compositional trends across the exoplanet mass range, and considerations for next-generation generalized retrieval frameworks. First, we present our systematic investigation of degeneracies between different model considerations in retrievals of transmission spectra and the observations that can resolve them. This study used a combination of Bayesian atmospheric retrievals and a range of common model assumptions, focusing on H2-rich atmospheres. We find that a combination of models including variable cloud coverage, prominent opacity sources, and high-precision optical and infrared spectra with current facilities enable constraints on cloud/haze properties and chemical abundances. Second, we apply our atmospheric retrieval framework to a large sample of 19 exoplanets ranging from cool mini-Neptunes to hot Jupiters. This effort constitutes the largest (i.e., broad wavelength coverage, multiple chemical species, mini-Neptunes to Jupiter sized planets) homogeneous chemical abundance survey for transiting exoplanets to date. We find a mass-metallicity trend of increasing H2O abundances with decreasing mass, significantly lower than the mass-metallicity relation for carbon in the solar system giant planets and similar predictions for exoplanets. On the other hand, the Na and K mass-metallicity trends are generally consistent with the solar system metallicity trend. We argue that the trends observed in this sample suggest different formation pathways for these close-in exoplanets compared to the long-period solar system giants. Third, we introduce Aurora, a next-generation retrieval framework for the characterization of H-rich and H-poor atmospheres. Here, we build upon state-of-the-art architectures and incorporate the following key advancements (a) a generalized compositional retrieval allowing for H-rich and H-poor atmospheres, (b) a generalized 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 then carry out an investigation of the current and future chemical composition constraints for exoplanet atmospheres using this new retrieval framework. We estimate the abundance constraints achievable for hot Jupiters, mini Neptunes, and rocky exoplanets with current and upcoming observational facilities. Lastly, we present our contribution to recent studies characterizing exoplanet atmospheres using ground and space-based facilities. We perform atmospheric retrievals on a diverse population of exoplanets from ultra-hot Jupiters to temperate mini Neptunes. Among the planets studied are WASP-127b, WASP-33b, WASP-21b, K2-18b, KELT-11b, and HAT-P-41b. Our results add to the vast chemical inventory of atomic and molecular species found in exoplanet atmospheres. Moreover, our analyses unveil some of the challenges when interpreting high-precision spectroscopic data and possible instrument systematics. The atmospheric reconnaissance presented in this work explores some of the considerations needed for generalized characterization of exoplanet atmospheres with upcoming ground-based and space-based facilities. We conclude this dissertation by summarizing our findings and their implications to the broader field of exoplanet characterization. We discuss some of the outstanding questions from our research and the prospect of future modeling and retrieval approaches to robustly characterize exoplanet atmospheres. The lessons from this work highlight that, although the inferences derived from observations are strongly influenced by model assumptions, the use of physically motivated models with minimal assumptions, and broadband transmission spectra with current and future facilities can provide plausible estimates for the atmospheric properties for planets outside our solar system.