A first-order theory is n-dependent if the edge relation of an infinite random n-hypergraph is not definable in any of its models. N-dependence is a strict hierarchy increasing with n, with the first level corresponding to the well-studied class of NIP theories. I will give a survey of recent work on n-dependent theories establishing connections to higher-arity generalizations of VC-dimension and hypergraph regularity (joint with Henry Towsner) and on understanding which algebraic structures are n-dependent (joint with Nadja Hempel).