This thesis confronts the challenges of complexity in financial systems, in which economic questions transform into computer science and mathematical problems. For instance, this occurs when systems have complicated dynamic interactions, and when systems are large and direct solution methods are not necessarily feasible. The first part of the thesis focuses on decentralized finance systems, which leverage new blockchain technologies to replace the role of risky financial intermediaries with decentralized (and more robust) structures, and the design of stablecoins, which aim to be a stable asset in a setting in which USD cannot be held directly. Decentralized stablecoins aim to function based on incentive design supported by transparent rules enforced cryptographically, in contrast to traditional currencies, which are supported by legal and governmental systems. This work explores new risks that complicate the design of resilient stablecoins, leading to significant results characterizing stablecoin runs and deleveraging spirals and new governance risks in decentralized finance protocols. The second part of this thesis focuses on cascades in economic networks, in which many firms interact with each other. This work explores new types of risks that can arise due to network structure and confronts two main issues that prevent the application of economic network models in practice, namely that these models can be very sensitive to parameter uncertainty and many aspects of these models can be computationally hard to compute. This leads to several significant results:- Characterizing mathematical properties of reinsurance contagion models, including dangerous structures that lead to retrocession spirals and underestimation of risk; - Adapting influence maximization methods to the problem of intervening in economic network contagion, enabling an NP-hard problem to be solved approximately in practice; - Applying perturbation theory to bound sensitivity in economic networks, improving on and unifying past results, and providing a means for network analysis to be more actionable in practice. This work involves applying methods from network analysis, perturbation theory, stochastic processes, agent-based models, game theory, and simulations.