In the first part I will recall the approach I, Adrian Iovita, Glenn Stevens and independently Vincent Pilloni have developed in order to define families of overconvergent ellitpic modular forms as sections of suitable line bundles. I will then report on work in progress with Iovita and Pilloni which allows to extend those constructions to the boundary of the weight space, providing characteristic p Banach modules with a compact operator. This confirms an expectation of Robert Coleman.