Pick an integer $n$. Consider a natural family of objects, such that each object $X$ in the family has an L-function $L(X,s)$. If we assume that the collection of special values $L^*(X,n)$ is bounded, does it imply that the family of objects is finite? We will first explain why we consider this question, in link with Kato's heights of mixed motives, and give two recent results: a Northcott property for families of Dedekind zeta functions, and a Northcott property for some families of L-functions attached to pure motives. We complete the picture with the most recent result, which concerns the same question for special values $L^*(X,1/2)$ in the case where 1/2 is the center of the critical strip. This is based on joint work with Riccardo Pengo, and on joint work with Jerson Caro and Riccardo Pengo.