We provide an overview of the relationship between financial networks and systemic risk. We present a taxonomy of different types of systemic risk, differentiating between direct externalities between financial organizations (e.g., defaults, correlated portfolios, fire sales), and perceptions and feedback effects (e.g., bank runs, credit freezes). We also discuss optimal regulation and bailouts, measurements of systemic risk and financial centrality, choices by banks regarding their portfolios and partnerships, and the changing nature of financial networks. Expected final online publication date for the Annual Review of Economics, Volume 13 is August 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

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We provide an overview of the relationship between financial networks and systemic risk. We present a taxonomy of different types of systemic risk, differentiating between direct externalities between financial organizations (e.g., defaults, correlated portfolios and firesales), and perceptions and feedback effects (e.g., bank runs, credit freezes). We also discuss optimal regulation and bailouts, measurements of systemic risk and financial centrality, choices by banks' regarding their portfolios and partnerships, and the changing nature of financial networks.

We analyze how interdependencies between organizations in financial networks can lead to multiple possible equilibrium outcomes. A multiplicity arises if and only if there exists a certain type of dependency cycle in the network that allows for selffulfilling chains of defaults. We provide necessary and sufficient conditions for banks' solvency in any equilibrium. Building on these conditions, we characterize the minimum bailout payments needed to ensure systemic solvency, as well as how solvency can be ensured by guaranteeing a specific set of debt payments. Bailout injections needed to eliminate self-fulfilling cycles of defaults (credit freezes) are fully recoverable, while those needed to prevent cascading defaults outside of cycles are not. We show that the minimum bailout problem is computationally hard, but provide an upper bound on optimal payments and show that the problem has intuitive solutions in specific network structures such as those with disjoint cycles or a core-periphery structure.