QCD at finite temperature and density on the lattice

In the first lecture we briefly summarize the basics of field theory thermodynamics and critical phenomena. We then introduce the lattice gauge field theory approach to QCD at finite temperature and density, which is a non-perturbative scheme allowing first 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 effective models of QCD. The second lecture is devoted to applications. We emphasize that current methods suffice to study the main phenomena at RHIC and LHC energies, and we discuss the ongoing theoretical efforts 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 field as of Summer 2008.