  We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type
theory for a class of symmetric jump processes(or equivalently, a class of
symmetric integro-differential operators). We focus on the sharp two-sided
estimates for the transition density functions (or heat kernels) of the
processes, a priori Holder estimate and parabolic Harnack inequalities for
their parabolic functions. In contrast to the second order elliptic
differential operator case, the methods to establish these properties for
symmetric integro-differential operators are mainly probabilistic.
