The theory of p-adic modular forms says that the classical modular form lives in p-adic families (for example, Hida's ordinary family, Coleman's overconvergent of finite slope family....). Then a natural question is the following: how the p-adic L function of modular form varies when the form varies in a p-adic family? In this talk, i will explain some results on this question using the deformations of Kato's Euler systems on different levels (over the weight space, over the cuspidal eigencurve and over the universal deformation rings) constructed by myself.