Although spatiotemporal graph neural networks have achieved great empirical success in handling multiple correlated time series, they may be impractical in some real-world scenarios due to a lack of sufficient high-quality training data. Furthermore, spatiotemporal graph neural networks lack theoretical interpretation. To address these issues, we put forth a novel mathematically designed framework to analyze spatiotemporal data. Our proposed spatiotemporal graph scattering transform (ST-GST) extends traditional scattering transforms to the spatiotemporal domain. It performs iterative applications of spatiotemporal graph wavelets and nonlinear activation functions, which can be viewed as a forward pass of spatiotemporal graph convolutional networks without training. Since all the filter coefficients in ST-GST are mathematically designed, it is promising for the real-world scenarios with limited training data, and also allows for a theoretical analysis, which shows that the proposed ST-GST is stable to small perturbations of input signals and structures. Finally, our experiments show that i) ST-GST outperforms spatiotemporal graph convolutional networks by an increase of 35% in accuracy for MSR Action3D dataset; ii) it is better and computationally more efficient to design the transform based on separable spatiotemporal graphs than the joint ones; and iii) the nonlinearity in ST-GST is critical to empirical performance. Processing and learning from spatiotemporal data have received increasing attention recently. Examples include: i) skeleton-based human action recognition based on a sequence of human poses (Liu et al. (2019)), which is critical to human behavior understanding (Borges et al. (2013)), and ii) multi-agent trajectory prediction (Hu et al. (2020)), which is critical to robotics and autonomous driving (Shalev-Shwartz et al. (2016)). A common pattern across these applications is that data evolves in both spatial and temporal domains.