My talk has two purposes. First, I will review the problems quantum tomography is designed to solve, its relation to other verification protocols, and the tools that have been proposed for this task. In a completely independent second part, I will report on recent work (around [arXiv:1712.08628]) on the representation theory of the Clifford group and its applications to property testing. In particular, I'll explain how "stabilizerness" and "Cliffordness" are properties of pure states and unitaries that can be tested from a system-size independent number of copies.