Higher-order graph neural networks (HOGNNs) and the related architectures from Topological Deep Learning are an important class of GNN models that harness polyadic relations between vertices beyond plain edges. They have been used to eliminate issues such as over-smoothing or over-squashing, to significantly enhance the accuracy of GNN predictions, to improve the expressiveness of GNN architectures, and for numerous other goals. A plethora of HOGNN models have been introduced, and they come with diverse neural architectures, and even with different notions of what the "higher-order" means. This richness makes it very challenging to appropriately analyze and compare HOGNN models, and to decide in what scenario to use specific ones. To alleviate this, we first design an in-depth taxonomy and a blueprint for HOGNNs. This facilitates designing models that maximize performance. Then, we use our taxonomy to analyze and compare the available HOGNN models. The outcomes of our analysis are synthesized in a set of insights that help to select the most beneficial GNN model in a given scenario, and a comprehensive list of challenges and opportunities for further research into more powerful HOGNNs.