In this talk I shall explain how the Kolyvagin systems associated to Beilinson-Kato elements for elliptic modular forms interpolate in the full deformation space (in particular, beyond the nearly-ordinary locus) and give rise to what I call a "universal Kolyvagin system". Along the way, I will indicate how we utilize these objects in order to define a quasicoherent sheaf on the Coleman-Mazur eigencurve that behaves like a p-adic L-function (in a certain sense of the word, in 3-variables). The universal Kolyvagin systems may be utilized so as to attempt a main conjecture over the eigencurve, which I will also talk about should time permit.