Numerical time integration of differential equations plays a pivotal role in computational sciences. This minisymposium focuses on the latest developments in time integrators, with a particular emphasis on exponential integration methods and parallelisation strategies used in such, with practical applications to test problems, thereby demonstrating their superior performance to conventional solvers. The session will showcase the novel ideas developed for treating the related exponential-like functions and their potential in leveraging high-performance computing systems for large-scale simulations. The audience will be acquainted with the state-of-the-art methods that enhance accuracy, efficiency, and fidelity of numerical simulations.