Beginning in the 80s with the celebrated work of Mazur, Tate and Teitelbaum, the study of exceptional zeros for p-adic L-functions has become a very fruitful area in number theory. One example is the recent proof of Gross' conjecture, which crucially relies on the theory of p-adic deformations of modular forms. In this talk, we give a historical survey of several applications of the theory of exceptional zeros, which incudes certain cases of the p-adic Birch and Swinnerton-Dyer conjecture and the Gross--Stark conjectures. We connect this with a recent result obtained in a joint work with V.Rotger, and which can be seen as a Gross--Stark formula for the adjoint of a weight one modular form. Finally, we take a glance to the theory of exceptional zeros from the point of view of Euler systems, exploring some tantalizing connections between the analytic and the algebraic world.