Abstract: The Dubrovin-Zhang (DZ) hierarchies and the Double Ramification (DR) hierarchies are two families of integrable hierarchies. In 2014, it was conjectured by Buryak that these hierarchies are equivalent. This talk is the first of a series of three talks about the proof of this conjecture, its refinements and related results. In the first talk we will present these hierarchies and explain how to reformulate this equivalence as a relation A=B of classes in the tautological ring of the moduli space of curves.