Advocates of cryptocurrencies such as Bitcoin believe that having currency competition will help achieve the economic objective of price stability. This Issue Brief summarizes research that explores whether competition among privately issued fiat currencies can actually produce price stability. The research finds that in most cases, a system of private monies does not deliver price stability. And even when it does, it always is subject to self-fulfilling inflationary episodes, and it supplies a suboptimal amount of money. Although there is no economic reason to curb the use of cryptocurrencies at the moment, it is important to review key regulatory issues that policymakers need to consider now, before the use of cryptocurrencies becomes even more widespread.