In recent years, spatio-temporal graph neural networks (GNNs) have attracted considerable interest in the field of time series analysis, due to their ability to capture dependencies among variables and across time points. The objective of the presented systematic literature review is hence to provide a comprehensive overview of the various modeling approaches and application domains of GNNs for time series classification and forecasting. A database search was conducted, and over 150 journal papers were selected for a detailed examination of the current state-of-the-art in the field. This examination is intended to offer to the reader a comprehensive collection of proposed models, links to related source code, available datasets, benchmark models, and fitting results. All this information is hoped to assist researchers in future studies. To the best of our knowledge, this is the first systematic literature review presenting a detailed comparison of the results of current spatio-temporal GNN models in different domains. In addition, in its final part this review discusses current limitations and challenges in the application of spatio-temporal GNNs, such as comparability, reproducibility, explainability, poor information capacity, and scalability.