A k-uniform hypergraph is said to be intersecting if no pair of edges is disjoint. The maximal size of an intersecting k-uniform hypergraph with a given groundset is given by the beautiful and well-known theorem of Erdos, Ko and Rado. A k-uniform hypergraph is s-almost intersecting if every edge is disjoint from exactly s other edges. Gerbner, Lemons, Palmer, Patkos and Szecsi made a conjecture on the maximal number of edges in such a hypergraph. We prove a strengthened version of this conjecture and determine the extremal graphs. We also give some related results and conjectures. Joint work with Elizabeth Wilmer.