The recent developments in the research area of stochastic partial differential equations (SPDEs) driven by Lévy noise and other non-Gaussian processes have significantly advanced the field. A notable trend is the exploration of numerical schemes that maintain high convergence orders across various norms, addressing the challenges posed by the non-Gaussian nature of the noise. Researchers are focusing on methods that can handle both the spatial and temporal aspects of SPDEs efficiently, often leveraging novel mathematical tools and adaptations of existing algorithms. For instance, the use of jump-adapted time discretization and the application of quantitative John--Nirenberg inequalities have shown promise in achieving uniform convergence orders. Additionally, the extension of Random Batch Methods to accommodate Lévy processes has demonstrated a substantial reduction in computational cost while preserving the dynamics of the original systems. These advancements are crucial for the accurate simulation of complex systems in fields such as fluid dynamics and nonlinear stochastic processes. Notably, the introduction of exponential stochastic Runge-Kutta methods for SPDEs of Nemytskii-type has opened new avenues for high-order convergence in the temporal domain, complementing existing spatial discretization techniques. These developments collectively push the boundaries of numerical analysis in SPDEs, offering more robust and efficient tools for researchers and practitioners in the field.