Coupled problems are ubiquitous across many applications in computational science and engineering. The coupling between different physical phenomena and spatio-temporal scales can give rise to strongly non-linear, non-convex, non-smooth, or highly ill-conditioned large-scale systems of equations, which require efficient coupling and solution strategies. In this minisymposium, we discuss the state-of-the-art and emerging trends in designing efficient and robust iterative methods for solving such systems of equations on modern computing architectures. We invite contributions related, but not limited to: - Preconditioning strategies - Domain decomposition and multilevel methods - Field-split methods - Acceleration techniques - Novel coupling strategies - Parallel algorithms and high-performance computing - AI enhanced numerical methods