Hypergraphs have seen widespread application in network and data science communities in recent years. Hypergraph structure in the form of hypernetwork science measures and hypergraph motifs have been among the common approaches. Meanwhile, there have also been theory advances in understanding hypergraph structure from a topological lens. In a recent survey article, my colleagues and I defined and described nine different homology theories for hypergraphs found in the literature based on constructing auxiliary structures---specifically simplicial, relative, and chain complexes. This talk will cover some highlights from this survey including how properties of some of the homology theories relate to hypergraph structural features, category theory approaches and functoriality of the auxiliary constructions, and illustrative examples to show variability and build intuition. I will also point out open questions in this area.