The Klein-Gordon equation was originally introduced as a relativistic model of the Schrodinger equation, but in some ways, its solutions exhibit behavior that is more like that of solutions to the wave equation. In this talk we will discuss recent results establishing upper bounds on the growth rate of finite time blowup solutions of the focusing nonlinear Klein-Gordon equation. All of the results that we will mention have analogues for the nonlinear wave equation. This is joint work with Rowan Killip and Monica Visan