This paper uses year-to-year variation in temperature to estimate the long-term effects of climate change on health outcomes in Mexico. Combining temperature data at the district level and three rounds of nationally representative household surveys, an individual's health as an adult is matched with the history of heat waves from birth to adulthood. A flexible econometric model is used to identify critical health periods with respect to temperature. It is shown that exposure to higher temperatures early in life has negative consequences on adult height. Most importantly, the effects are concentrated at the times where children experience growth spurts: infancy and adolescence. The robustness of these findings is confirmed when using health outcomes derived from accidents, which are uncorrelated with early exposure to high temperatures.