This dissertation is comprised of three parts, with each part focusing on topics in axions anddark matter, anomaly cancellation in supergravity theories, and the implications of the string swampland to cosmology, respectively. In Part I, we consider axions in particle extensions of the Standard Model and models of dark matter. We present a class of models where supersymmetry and the Peccei–Quinn symmetry are simultaneously broken and the messengers that mediate the effects of these symmetry breakings to the Standard Model are identical. We also describe a production scenario for QCD axion dark matter where the Peccei-Quinn phase transition occurs at a temperature far below the symmetry breaking scale. The produced axions tend to be warm. For a certain range of the decay constant, the effect of the predicted warmness on structure formation can be confirmed by future observations of 21 cm lines. Additionally, a portion of parameter space requires a mixing between the Peccei-Quinn symmetry breaking field and the Standard Model Higgs and thereby predicts an observable rate of rare Kaon decays. We also consider the late universe cosmology of ultralight axion dark matter models, and show that requiring the axion to have a matter-power spectrum that matches that of cold dark matter constrains the magnitude of the axion couplings to the visible sector. Comparing these limits to current and future experimental efforts, we find that many searches require axions with an abnormally large coupling to Standard Model fields, independently of how the axion was populated in the early Universe. We survey mechanisms that can alleviate the bounds, namely, the introduction of large charges, various forms of kinetic mixing, a clockwork structure, and imposing a discrete symmetry. We provide an explicit model for each case and explore their phenomenology and viability to produce detectable ultralight axion dark matter. In Part II, we use Pauli–Villars regularization to evaluate the conformal and chiral anomalies in the effective field theories from Z3 and Z7 compactifications of the heterotic string without Wilson lines and a Z3 compactification of the heterotic string with two Wilson lines and an anomalous U(1). We show that parameters for Pauli–Villars chiral multiplets can be chosen in such a way that the anomaly is universal in the sense that its coefficient depends only on a single holomorphic function of the three diagonal moduli. It is therefore possible to cancel the anomaly by a generalization of the four-dimensional Green–Schwarz mechanism. In particular, we are able to reproduce the results of a string calculation of the four-dimensional chiral anomaly for these models. In Part III, we discuss the relations between swampland conjectures and observational constraints on both inflation and dark energy. Using the requirement |∇V | ≥ cV , with c as a universal constant whose value can be derived from inflation, there may be no observable distinction between constant and nonconstant models of dark energy. However, the latest modification of the above conjecture, which utilizes the second derivative of the potential, opens up the opportunity for observations to determine if the dark energy equation of state deviates from that of a cosmological constant. We also comment on the observability of tensor fluctuations despite the conjecture that field excursions are smaller than the Planck scale.