In this talk, we first discuss approximate factorizations of heat kernels and Green functions for purely discontinuous Markov processes. Under natural conditions, we show that the approximate factorization of the heat kernel is equivalent to that of the Green function. In the second part, we will discuss applications of these factorizations to derive two-sided heat kernel estimates for three classes of processes: stable-like processes with critical killing in $C^{1,Dini}$ open sets; killed stable-like processes with low regularity coefficients; and non-symmetric stable processes in $C^{1,2-Dini}$ open sets. In particular, we obtain sharp, explicit two-sided estimates for the killed and censored stable processes in $C^{1,Dini}$ open sets. This is based on joint work with Professor Renming Song (UIUC).