We intend to show some relations between several types of inverse problems and controllability methods and results. Actually we will present some results concerning data assimilation with a quite novel point of view for parabolic problems first and then for hyperbolic problems. In a second step, we will apply this to the question of source recovery for hyperbolic equations from boundary measurements and we will also show the limits of this question with some counterexamples. In a third part we will present some results concerning the recovery of a potential in a hyperbolic equation (or a Schrödinger equation) from boundary measurements and we will present some important open problems.