The transport and spectral properties of heavy quarkonia in hot QCD matter are a central ingredient to describe their observables in high-energy heavy-ion collisions. We review recent activity in evaluating these properties, including a nonperturbative quantum many-body approach where the basic two-body interaction kernel is constrained by quantities that can be computed with good precision in thermal lattice QCD. We then give a brief overview of quarkonium transport approaches to heavy-ion collisions. Focusing on the semiclassical approach we discuss the current interpretation of charmonium and bottomonium observables at RHIC and the LHC, including excitation functions that started with 𝐽/𝜓 and 𝜓′ data from the heavy-ion program at the SPS.