A free boundary minimal surface (FBMS) in a given three-dimensional Riemannian manifold is a critical point of the area functional with respect to variations that constrain its boundary to the boundary of the ambient manifold.

We will talk about the existence of FBMS, with particular emphasis on an equivariant min-max method and on the computation of the Morse index (i.e., the number of directions in which the area decreases at second order) of the constructed FBMS. Moreover, we will review some existing results relating the Morse index of a FBMS and investigate some further developments.