This thesis studies the origins and consequences of financial crises, and computational techniques to solve continuous-time economic models that explain such crises. The first chapter shows that financial recessions are typically characterised by a large risk premium and a slow recovery. However, macro-finance models have trouble matching these empirical features, especially when they are calibrated to match both the observed unconditional and conditional macroeconomic and asset-pricing moments simultaneously. In this chapter, I build a macro-finance model that quantitatively explains the salient features of a financial crisis, such as a large drop in output, a spike in the risk premium, reduced financial intermediation, and a long duration of economic distress. The model has leveraged intermediaries with stochastic productivity and a state-dependent exit rate that governs the transition into and out of a crisis. A model without these two features suffers from a trade-off between the amplification and persistence of a crisis. I show that my model resolves this tension and generates realistic crisis dynamics. In the second chapter, I develop a new computational framework called Actively Learned and Informed Equilibrium Nets (ALIENs) to solve continuous time economic models with endogenous state variables and highly non-linear policy functions. I employ neural networks that are trained to solve supervised learning problems that respect the laws governed by the economic system in the form of general parabolic partial differential equations. The sub-domain of the high dimensional state space that carries the most economic information is learned actively in an iterative loop, enforcing the random training points to be sampled from areas that matter the most to ensure convergence. The method is applied to successfully solve a model of macro-finance that is notoriously difficult to handle using traditional finite difference schemes. In the third chapter, I investigate the origins of bank failures, an important feature of financial crises. I analyze a panel of bank holding companies and offer empirical evidence for the franchise value to be associated with a higher probability of failure. The results indicate that the changing scope of the banking industry with declining franchise value compared to the pre-crisis period is worrisome, despite strong capital ratios.