A reaction network is a graphical configuration that can describe many biochemical systems with interactions between species (molecules). If the abundances of the species involved in a reaction network are small, then the randomness inherent in the molecular interactions is important to the system dynamics, and hence the abundances are modeled stochastically with a jump-by-jump fashion continuous-time Markov chain. In the first half of this talk, we discuss classical and recent issues in reaction network literature, which especially include how the graphical structures of reaction networks are related to stability and mixing times of the associated Markov processes. We will also discuss how the stability and mixing times can be used for practical problems in system biology and biology computing.