In a beautiful 1969 paper, Yuri Manin proved that for any number field $K$ and any prime number $p$, there is an absolute bound on the order of any $p$-power torsion point on any elliptic curve over $K$. Although now superseded by theorems of Mazur and Merel, Manin's result has a particularly simple proof, relying only on the theory of heights and some facts about modular curves. This talk will explain Manin's proof.