  In these lecture notes we will discuss recent progress in extracting spectral
and transport properties from lattice QCD. We will focus on results of probes
of the thermal QCD medium as well as transport coefficients which are important
ingredients for hydrodynamic and transport models that describe the evolution
of the produced medium. These include electromagnetic probes, like the rates of
emitted photons and dileptons, quarkonium spectral functions as well as
transport coefficients like the electrical conductivity or heavy flavor
diffusion coefficients of the quark gluon plasma (QGP). All these real time
quantities are encoded in the vector meson spectral function. A direct
determination of the spectral functions is not possible in Euclidean lattice
QCD calculations but they can be analytically continued from imaginary to real
time. Therefore it is possible to relate the spectral function to the
corresponding Euclidean correlation functions. In the following sections we
will discuss the procedure to determine the required correlation functions and
the extraction of the spectral functions from lattice QCD correlators. We will
illustrate the concepts and methods to obtain spectral functions and related
physical observables from continuum extrapolated correlation functions. We will
focus here on results obtained from continuum extrapolated lattice correlation
functions, which requires large and fine lattices, which so far was only
possible in quenched approximation. We will only give a brief introduction to
lattice QCD and refer to the textbooks [1,2,3,4] and lecture notes [5] for more
detailed introductions to lattice field theory. For the topics addressed in
this lecture note we also like to refer to the overview articles on QCD
thermodynamics and the QCD phase transition [5,6,7] and quarkonium in extreme
conditions [8].
