I will review various presentations of the elliptic Hall algebra by generators and relations, including the recently discovered relation with the double Dyck path algebra of Carlsson and Mellit. I will then explain how these generators appear in various categorical setups such as the derived category of the commuting stack and the trace of the affine Hecke category, and state several conjectures about them. The talk is based on joint works with Andrei Negut, Nicolle Gonzalez and Jose Simental.