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.