**Abstract:** In this talk, we focus on the set SNA(M, Y) of those Lipschitz maps from a complete metric space M to a Banach space Y which strongly attain their Lipschitz norm (i.e. the supremum defining the Lipschitz norm is a maximum). We prove that this set is not norm-dense when M is a length space, when M is a closed subset of R with positive Lebesgue measure, and when M is the unit circle. This provides new examples which have very different topological properties than the previously known ones. In addition, we study the linear properties which are sufficient to get Lindenstrauss property A for the Lipschitz-free space over M, and show that all of them actually provide the norm-density of SNA(M, Y). This is part of a joint work with B. Cascales, R. Chiclana, M. Martín and A. Rueda Zoca.