Double saddle-point systems are drawing increasing attention in the past few years, due to the importance of relevant applications and the challenge in developing efficient numerical solvers. In this talk we describe some of their numerical properties and make distinctions between this family and standard saddle-point systems. We derive eigenvalue bounds, expressed in terms of extremal eigenvalues and singular values of block sub-matrices. The analysis includes bounds on preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown that in this case the eigenvalues are clustered within a few intervals bounded away from zero, giving rise to rapid convergence of Krylov subspace solvers. A few numerical experiments illustrate our findings.