The ongoing substance misuse epidemic in the United States is complex and dynamic and should be approached as such in the development and evaluation of policy1. Drug overdose deaths (largely attributable to opioid misuse) in the United States have grown exponentially for almost four decades, but the mechanisms of this growth are poorly understood2. From analysis of 661,565 overdose deaths from 1999 to 2017, we show that the age-specific drug overdose mortality curve for each birth-year cohort rises and falls according to a Gaussian-shaped curve. The ascending portion of each successive birth-year cohort mortality curve is accelerated compared with that of all preceding birth-year cohorts. This acceleration can be attributed to either of two distinct processes: a stable peak age, with an increasing amplitude of mortality rate curves from one birth-year cohort to the next; or a youthward shift in the peak age of the mortality rate curves. The overdose epidemic emerged and increased in amplitude among the 1945-1964 cohort (Baby Boomers), shifted youthward among the 1965-1980 cohort (Generation X), and then resumed the pattern of increasing amplitude in the 1981-1990 Millennials. These shifting age and generational patterns are likely to be driven by socioeconomic factors and drug availability, the understanding of which is important for the development of effective overdose prevention measures.