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In the rst lecture we briey summarize the basics of eld theory thermodynamics  and critical phenomena. We then introduce the lattice gauge eld theory approach  to QCD at nite temperature and density, which is a non-perturbative scheme  allowing rst principle calculations using the QCD Lagrangian as a sole input.  Some of the general concepts and idea introduced at the beginning are  demonstrated by use of simple eective models of QCD. The second lecture is  devoted to applications. We emphasize that current methods suce to study the  main phenomena at RHIC and LHC energies, and we discuss the ongoing theoretical  eorts devoted to the solution of the sign problem which hampers the simulations  of cold and dense matter. We conclude with short overview of the status of the  eld as of Summer 2008

QCD at ﬁnite temperature and density on the lattice