A fundamental endeavour in macromolecular science is the control of molecular-level complexity, including molecular weight distribution, end groups and architecture. Since the discovery that native biomacromolecules can have a specific sequence translating in a specific biological function, controlling individual monomer sequence has become the ultimate expression of molecular-level complexity. Replicating this remarkable structural precision in abiological macromolecules has emerged as a defining goal and challenge within polymer science. In this Review, we survey developments in synthetic methods, characterisation techniques, simulation workflows and applications relevant to this goal. We also address the broader question of to what extent is such control of molecular-level complexity significant in macromolecules. Specifically, we will focus on characterisation in this Review because of its importance in connecting synthesis with applications.