Physics seldom makes breakthroughs when we consider objects only in isolation. Instead we model and measure how objects sit among their neighbours, spatial, temporal, or otherwise: a galaxy in its environment, a halo in its history, a photon hit in a detector. Graphs are ideally suited for encoding exactly these relations, and Graph Neural Networks (GNNs) learn the geometric functions that respect them. This review presents the use of GNNs for astrophysics and cosmology, and argues that GNNs often suit physics and physical data better than any alternative: they fit the native geometry of the domain and the signal, enable the incorporation of the symmetries and prior knowledge we already trust and have toiled to build, and stay flexible enough to capture what we do not yet understand. GNNs' physically intuitive workings also render them more interpretable than other deep-learning architectures. We show how we can begin from general principles of the geometry of deep learning and build the graph and the graph neural network from the ground up, and how other architectures, like the convolutional network and the transformer, can be seen as special cases of one unified geometric blueprint. We survey applications across cosmology, galaxy formation, dynamics, and instrumentation; and we chart the frontier. From this survey, we find that only one in six articles outside instrumentation applies GNNs to data rather than just simulations. We also distil a set of best practices, including when not to reach for a graph. Throughout, we will prioritise physical intuition and foundational principles over mathematical machinery whenever possible.

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