Rising healthcare costs have sparked debate about the best way to provide high-quality affordable health insurance. We discuss the potential for regulated insurance markets to outperform single-payer public insurance. We use as an example the private Medicare plans that now provide insurance to almost a third of seniors in the United States. The evidence suggests that private plans can limit costs and potentially appeal to enrollees, and that well-designed risk adjustment can mitigate market failures due to adverse selection. However, fostering competition between insurers, especially in smaller markets, is difficult. We discuss how future research might illuminate the relative advantages of public and private health insurance.