Modern discretization and solution methods for time-dependent PDEs consider the full problem in space and time simultaneously and aim to overcome limitations of classical approaches by first discretizing in space and then solving the resulting ODE, or first discretizing in time and then solving the PDE in space. The development of space-time methods for hyperbolic and parabolic differential equation is an emerging and rapidly growing field in numerical analysis and scientific computing. At the first Workshop on this topic in 2017 a large variety of interesting and challenging concepts, methods, and research directions have been presented; now we exchange the new developments. The focus is on the optimal convergence of discretizations and on efficient error control for space-time methods for hyperbolic and parabolic problems, and on solution methods with optimal complexity. This is complemented by applications in the field of time-dependent stochastic PDEs, non-local material laws in space and time, optimization with time-dependent PDE constraints, and multiscale methods for time-dependent PDEs.
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