Iterative methods lie at the heart of modern scientific computing, providing efficient and scalable solvers for large-scale systems arising from the discretization of partial differential equations (PDEs) and related optimization problems. When solving such problems numerically, the complexity of the underlying phenomena, such as multiphysics formulation, high-frequency wave propagation, and long time scale, often requires extremely fine space and/or time discretizations, which leads to very large systems. Iterative solvers are among the most efficient methods to tackle this type of problem. This minisymposium focuses on recent developments in the design, analysis, and application of new iterative methods for both linear and non-linear problems. The talks will cover advances in classical frameworks such as domain decomposition, parallel-in-time methods, as well as novel preconditioning strategies and convergence acceleration techniques. Particular attention will be given to non-linear extensions, including modulus-based and fixed-point type iterations, where non-linearity introduces new analytical and computational challenges. The invited speakers will present theoretical insights into convergence behavior and robustness of iterative solvers, as well as algorithmic innovations adapted to parallel and time-dependent settings. The overall goal is to bring together experts working on different classes of iterative methods to exchange ideas and discuss associated open challenges in this research field.