A finite connected acyclic graph is called a tree. Both properties - connectivity and being acyclic make very good sense in higher dimensions as well. This has led Gil Kalai (1983) to define the notion of a $d$-dimensional hypertree for $d>1$. The study of hypertrees is an exciting area of research, and I will try to give you a taste of the many open questions and what we know (and do not know) about them. No specific prior background is assumed.
The talk is based on several papers. The list of coauthors on these papers includes Roy Meshulam, Mishael Rosenthal, Yuval Peled, Ilan Newman and Yuri Rabinovich.