Nonlinear dynamical systems are ubiquitous in both engineering and the physical sciences. They are characterised by complex behaviours such as multi-stability, bifurcations, and chaos. Continuation methods have emerged as powerful tools for exploring the various solutions of dynamical systems as parameters are varied. These techniques enable the computation and tracking of equilibria, periodic and quasi-periodic responses, as well as bifurcation points, allowing for the identification of emerging solution branches and critical thresholds where the system exhibits both qualitative and quantitative changes. This minisymposium aims to bring together researchers and practitioners working on recent developments in continuation methods and their applications to nonlinear dynamics. We welcome contributions that advance the theoretical foundations, numerical algorithms, and computational tools for continuation analysis, as well as applications in solid and fluid mechanics, multiphysics systems such as fluid-structure interactions or electromechanical systems, biological models, and related areas.