Evolutionary Partial Differential Equations (PDEs) are of primary relevance: they arise pervasively in nature and, consequently, they are present in a wide range of problems in engineering and applied physics. The numerical approximation of their solutions is a challenging task that has always attracted interest within the scientific community. This minisymposium is devoted to novel and advanced high-order space--time methods for computing accurate and efficient numerical approximations of the solutions of evolutionary PDEs. Its main goal is to bring together experts in numerical analysis from different international research groups to discuss the most recent advances and the open challenges. Topics of interest include, but are not limited to, conforming discretizations, least-squares formulations, $hp$-convergence, a priori and a posteriori error estimates, stability, structure-preserving methods, optimal control, and isogeometric discretizations, with applications to parabolic, hyperbolic, and dispersive problems.