Arms races present a major threat to global stability, and in the nuclear age this entails a significant threat to humanity as a whole. However, arms races are not a recent phenomenon, and some literature has been built up in terms of understanding the mechanics and dynamics of arms races. The first mathematical model of arms races was developed between 1920 and 1960 by Lewis Richardson, and this model inspired a distinctly different model from Kenneth Boulding. I will be contrasting the Richardson and Boulding models of arms races, first explaining how each works and then applying both to two separate, prominent arms races: the pre-WWI naval arms race between Germany and Britain and the Cold War arms race between the U.S. and the U.S.S.R. I will critique both models, drawing out what they successfully capture and what they fail to capture.