Feedback loops are common in systems such as gene regulation and protein signaling, yet most causal discovery methods assume that the underlying relationships are acyclic. This dissertation develops methods for learning structural causal models with feedback loops and characterizes what can be identified when measurements are incomplete, noisy, or affected by unmeasured confounding. For systems observed at equilibrium, it introduces four likelihood-based methods that combine observational and interventional data to learn causal graphs, allowing for nonlinear interactions and accounting for latent confounders, missing values that may depend on unobserved data, and measurement noise. It establishes conditions under which these methods recover the true graph up to an equivalence class. The dissertation also studies how to design sparse measurements that preserve information about causal structure in linear acyclic systems when there are fewer measurements than variables. Finally, for continuous-time systems with unmeasured confounding, it identifies conditions under which observed trajectories determine the system features needed to predict the effects of interventions.