I discuss the econometrics and the economics of past research on the effects of minimum wages on employment in the United States. My intent is to try to identify key questions raised in the recent literature, and some from the earlier literature, which I think hold the most promise for understanding the conflicting evidence and arriving at a more definitive answer about the employment effects of minimum wages. My secondary goal is to discuss how we can narrow the range of uncertainty about the likely effects of the large minimum wage increases becoming more prevalent in the United States. I discuss some insights from both theory and past evidence that may be informative about the effects of high minimum wages, and try to emphasize what research can be done now and in the near future to provide useful evidence to policymakers on the results of the coming high minimum wage experiment, whether in the United States or in other countries.