There are natural connections between transient dynamics (such as transient chaos and transient oscillations) of a complex system and a quasi-stationary distribution (QSD) of the reduced stochastic system. QSD are those almost invariant to a diffusion process over exponentially long time, representing important transient stochastic dynamics. They arise frequently in applications especially in chemical reactions and population systems admitting extinction states. This talk will present some rigorous results on the existence, uniqueness, concentration, and convergence of QSDs along with their connections to the spectrum of the Fokker-Planck operator.