The Boltzmann hierarchy is an infinite system of coupled equations that are instrumental for the rigorous derivation of the Boltzmann equation from many particles. In this talk we will first show uniqueness of the mild solutions to the Boltzmann hierarchy by combining, for the first time, a combinatorial technique known as the Klainerman-Machedon board game argument together with an $L^\infty$-based estimate. Then we will show existence of global in time mild solutions to the Boltzmann hierarchy for admissible initial data. The proof of existence is constructive, and employs known global in time solutions to the Boltzmann equation via a Hewitt-Savage type theorem.