A central counterparty (CCP) is a financial market utility that lowers counterparty default risk on specified financial contracts by acting as a buyer to every seller, and as a seller to every buyer. When at risk of failure, a CCP could be forced into a normal insolvency process such as bankruptcy, or an administrative failure resolution process. This chapter reviews some alternative approaches to the design of insolvency and failure resolution regimes for CCPs. I focus on the allocation of losses and the question of whether and how to provide for continuity of clearing services. I discuss how one might adapt to CCPs some of the failure resolution approaches currently being designed for other forms of systemically important financial institutions. A key policy question is when to interrupt a contractually based CCP default management process with an overriding failure resolution process.