This minisymposium focuses on recent advances in space-time finite element methods and high-order implicit time integration for the numerical solution of time-dependent partial differential equations. High-order space-time and implicit methods pose significant challenges. Solving the arising large coupled systems demands new algorithmic approaches and computationally efficient implementations. This minisymposium brings together contributions addressing these challenges through advanced discretization techniques, efficient solvers, and high-performance implementations. A central focus is the interplay between high-order discretizations and the scalable solvers that make them practical. Talks will cover the design and analysis of multigrid and multilevel solvers tailored to space-time finite element frameworks. These include geometric and algebraic multigrid approaches, domain decomposition methods, and parallel-in-time algorithms that overcome the limitations of sequential time stepping and enable strong scalability on modern hardware. Another major focus is on robust block preconditioners for monolithic space-time systems. These preconditioners are essential for achieving optimal solver performance independent of the mesh resolution and polynomial degree. Recent advances in block-structured approaches have demonstrated significant gains in robustness and efficiency and established them as key building blocks for the development of scalable and reliable solvers for space-time systems. An emphasis is on space-time tensor-product techniques, variational time integrators, and structure-preserving discretizations. Contributions include novel Galerkin methods and formulations tailored to stiff or multiscale problems. The translation of theoretical advances into practical performance requires hardware-aware implementations that minimize data movement and memory usage. Contributions will address matrix-free operator application, hardware-aware algorithm design, and cache-optimized and performance-portable implementations. The minisymposium aims to foster exchange across numerical mathematics and scientific computing. The overarching goal is to advance the theoretical understanding, algorithmic design, and computational realization of space-time finite element methods and implicit high-order time integration as scalable and efficient tools for the simulation of complex dynamical systems.