Topological deep learning is a rapidly growing field focused on developing deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize graphs and many other domains encountered in scientific computations. In this talk, Tolga will present a unifying deep learning framework built upon these rich data structures of higher-order relationships. Specifically, he will begin by introducing the field of topological deep learning and its potential applications and challenges. He will then dive into a novel type of topological domain: combinatorial complexes. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations, while permitting the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. In this way, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces. Tolga will then develop a general class of message-passing combinatorial complex neural networks (CCNNs), focusing primarily on attention-based CCNNs. He will additionally touch on the permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs. The performance of CCNNs on tasks related to mesh shape analysis and graph learning will be provided. The experiments demonstrate that CCNNs have competitive performance compared to state-of-the-art deep learning models specifically tailored to the same tasks. These findings demonstrate the advantages of incorporating higher-order relations into deep learning models and show great promise for AI4Science.