Saddle-point systems lie at the heart of many numerical formulations of incompressible fluid flow, where they naturally arise from pressure-velocity coupling in the Navier–Stokes equations and from constaint variational formulations. Efficient and robust solution strategies for such systems are crucial in a wide range of applications, from direct numerical simulations of turbulent flows to fluid–structure interaction and multiphysics coupling. Saddle-point systems pose significant numerical challenges: they are typically large, sparse, indefinite, and sensitive to discretization choices, boundary conditions, and physical parameters. This minisymposium highlights recent advances in the numerical solution of such systems, with emphasis on techniques motivated by or applied to fluid flow problems. Contributions span a spectrum of perspectives — from the development of robust and efficient preconditioners and iterative solvers to the design of coupling strategies within fluid–structure interaction and immersed boundary frameworks. We aim to foster fruitful exchange between numerical analysts, applied mathematicians, and computational scientists working on fluid flow problems. This minisymposium will serve as a platform for sharing insights, identifying common challenges, and exploring promising directions for future research in the numerical solution of saddle-point systems.