In auction and mechanism design, Myerson's classical regularity condition is often too weak for a quantitative analysis of performance. For instance, ratios between revenue and welfare, or sales probabilities may vanish at the boundary of Myerson regularity. This paper introduces Lambda-regularity as a quantitative measure of how regular a distribution is. Lambda-regularity includes Myerson regularity and the monotone hazard rate condition as special cases. We show that Lambda-regularity implies sharp bounds on various key quantities in auction theory, thus extending several recent findings from quantitative auction and mechanism design.