given an elliptic curve e, kolyvagin used cm points on modular curves to construct a system of classes valued in the galois cohomology of the torsion points of e. under the conjecture that not all of these classes vanish, he gave a description for the selmer group of e. this talk will report on recent work proving new cases of kolyvagin's conjecture. the methods follow in the footsteps of wei zhang, who used congruences between modular forms to prove kolyvagin's conjecture under some technical hypotheses. we remove many of these hypotheses by considering congruences modulo higher powers of p. the talk will explain the difficulties associated with higher congruences of modular forms and how they can be overcome. i will also provide an introduction to the conjecture and its consequences, including a `converse theorem': algebraic rank one implies analytic rank one.