Graphs are ubiquitous in nature and can serve as models for many applied or theoretical problems. Thus, graph-based methods in machine learning can be applied to a wide variety of problems and applications. In this talk, I will provide a brief introduction to the basics of graph neural networks (GNNs) and then discuss several topics from my group's work. This includes the expressivity of GNNS, i.e., which functions on graphs can GNNs approximate and which graphs or nodes within a graph can they distinguish? A GNN algorithm considering the conservation law inherent in graphs associated with a flow of physical resources. As well as GNN applications in the power grid, and for the automatic optimization of printed circuit boards.