  A general introduction into the subject aimed at a general theoretical
physics audience. We introduce the sign problem posed by finite density lattice
QCD, and we discuss the main methods proposed to circumvent it, with emphasis
on the imaginary chemical potential approach. The interrelation between the
Taylor expansion and the analytic continuation from imaginary chemical
potential is discussed in detail. The main applications to the calculation of
the critical line, and to the thermodynamics of the hot and the hadronic phase
are reviewed.
