Optimization and inverse problems are ubiquitous in applied mathematics, with applications ranging from engineering and physics to imaging and data science. The complexity of these phenomena, such as multiphysics formulation, high frequency wave propagation, and long timescale, often necessitates advance tailored discretizations, which eventually lead to very large systems and require using efficient numerical techniques. Solvers both iterative and direct play a crucial role in addressing these problems. This minisymposium explores recent ad- vances and emerging trends in the development of these solvers, discussing both theoretical insights, numerical implementations and practical applications.