The bound states of a heavy quark and antiquark ($c\bar{c}, b\bar{b}$) are ideal probes to explore the quark-gluon plasma created in relativistic heavy-ion collisions at the RHIC and LHC. Not only have they become experimentally accessible with high precision but also efficient tools, so called effective field theories (EFT) have been developed to treat them theoretically. Here we present recent progress in understanding the in-medium behavior of heavy-quarkonium with the help of EFT's combined with non-perturbative and first principles simulations in lattice QCD. In particular we discuss computations of heavy quarkonium spectral functions with the help of Bayesian unfolding methods and the physics we can extract from them. Limitations and the underlying assumptions of the used approaches are pointed out.