Sets of uniqueness and their properties are traditionally investigated in Harmonic Analysis. The study of such sets has a long and illustrious history, witnessing fruitful interdisciplinary interactions, often enriching the subject with a vibrant fertility for crossover of ideas. In this talk, we set up the modern framework used to study such sets with particular emphasis on some (classical) descriptive set-theoretic aspects. We present some results concerning the family of closed sets of uniqueness of a locally compact Polish group - more concretely, we will talk about their complexity and the (in)existence of a Borel basis.