In 1993, Haagerup posed the question 'what are the finite depth subfactors with index slightly larger than 4?' He came up with a list of candidate principal graphs. Most of these were ruled out using fusion category (Bisch) or number-theoretic (Asaeda-Yasuda) techniques. Two of the remaining graphs correspond to previously unknown subfactors (Asaeda-Haagerup). The speakers and Scott Morrison recently constructed a subfactor for the last remaining graph on Haagerup's list, which we call 'extended Haagerup'. We will frame this talk with classification results, beginning with Haagerup's and ending with a rapid summary of work in progress (by subsets of Calegari, Jones, Morrison, Penneys, Peters, Snyder) on the classification of subfactors of index less than 5. The majority of this talk, however, will focus on a uniform generators-and-relations construction of the Haagerup and extended Haagerup planar algebras, and their properties. Emily will begin with general results on planar algebra constructions, exploiting the 'annular structure' in order to get generators with nice properties. Stephen will discuss further structure results including a complete list of relations, a basis for the box spaces, and the 'jellyfish algorithm' for evaluating closed diagrams. Finally, Noah will give an application of this description to the noncyclotomicity of some associated fusion categories.