Topological Neural Networks (TNNs) are deep learning architectures that process signals defined on topological domains, such as hypergraphs and cellular complexes -- hence generalizing Graph Neural Networks. The additional flexibility and expressivity of TNN architectures permits the representation and processing of complex natural systems such as proteins, neural activity, and many-body physical systems. This talk synthesizes the recent TNN literature using a single unifying notation and graphical summaries and sheds light on existing challenges and exciting opportunities for future development.