  This review gives a critical assessment of the current state of lattice
simulations of QCD thermodynamics and what it teaches us about hot hadronic
matter. It outlines briefly lattice methods for studying QCD at nonzero
temperature and zero baryon number density with particular emphasis on
assessing and reducing cutoff effects. It discusses a variety of difficulties
with methods for determining the transition temperature. It uses results
reported recently in the literature and at this conference for illustration,
especially those from a major study carried out by the HotQCD collaboration.
