This article addresses the problem of comparing abilities of players from different eras in professional sports. We study National Hockey League players, professional golfers, and Major League Baseball players from the perspectives of home run hitting and hitting for average. Within each sport, the careers of the players overlap to some extent. This network of overlaps, or bridges, is used to compare players whose careers took place in different eras. The goal is not to judge players relative to their contemporaries, but rather to compare all players directly. Hence the model that we use is a statistical time machine. We use additive models to estimate the innate ability of players, the effects of aging on performance, and the relative difficulty of each year within a sport. We measure each of these effects separated from the others. We use hierarchical models to model the distribution of players and specify separate distributions for each decade, thus allowing the “talent pool” within each sport to change. We study the changing talent pool in each sport and address Gould's conjecture about the way in which populations change. Nonparametric aging functions allow us to estimate the league-wide average aging function. Hierarchical random curves allow for individuals to age differently from the average of athletes in that sport. We characterize players by their career profile rather than a one-number summary of their career.