Some of the most beautiful results in mechanism design depend crucially on Myerson's (1981) regularity condition. E.g., the second-price auction with reserve price is revenue maximizing only if the type distribution is regular. This paper offers two main results. First, an interpretation of regularity is developed in terms of being the next to fail. Second, using expanded concepts of concavity, a tight sufficient condition on the density function is formulated. New examples of parameterized distributions are shown to be regular. Applications include standard design problems, optimal reserve prices, the analysis of bidding data, and multidimensional types.