Cluster algebras were introduced by Fomin and Zelevinsky in 2002. They are subalgebras of a Laurent polynomial ring, defined in terms of combinatorial data (quivers and mutations). Cluster algebras have quickly gained wide interest in the community and have appeared in many contexts. In this talk we view them from the perspective of an age old question in this new setting: when do elements of a cluster algebra possess unique factorizations into prime elements?

This is joint work with Ana Garcia Elsener and Philipp Lampe. arXiv:1712.06512.