Heegner points are special points on elliptic curves constructed from CM points on a modular parametrization. In this talk I will describe this construction and sketch a proof that Heegner points corresponding to distinct CM fields are linearly independent in the Mordell-Weil group. This has a negative implication to ECDLP, because if such Heegner points had been dependent, then ECDLP might have been solvable via Deuring-Heegner lifts. (Joint work with Michael Rosen)