Stochastic dynamical systems with absorbing states are used to model systems arising from ecology, biology, chemical kinetics, and other fields. Despite the fact that these systems are eventually absorbed, they often persist for long periods of time prior to absorption. Quasi-stationary distributions (QSD) are the fundamental mathematical objects used to characterize the stability and long-term behavior of such systems prior to absorption. In the first part of this talk, I will consider a collection of Markov chains that model the evolution of multi-type biological populations. The state space of the chains is the positive orthant, with absorption at the boundary of the orthant, which represents the extinction of different population types. The main results discussed in this part of the talk show that, as the size of the system increases, the behavior of the associated QSD can be characterized in terms of an underlying continuous-time dynamical system. The proofs of these results rely on uniform large deviation results for small noise stochastic dynamical systems and methods from the theory of dynamical systems. In the second part of this talk, I will introduce two stochastic approximation schemes that can be used to estimate the QSD of a finite-state Markov chain with absorbing states. Both methods are described in terms of a collection of particles evolving via interacting chains in which the interaction is given in terms of the total time occupation measure of all particles in the system and has the impact of reinforcing certain types of transitions. Under the key assumption that the ratio between the number of particles in the system and time goes to zero, I will discuss the asymptotic behavior of these approximation methods as time and the number of particles in the system simultaneously become large. This talk is based on joint work with Prof. Amarjit Budhiraja and Prof. Nicolas Fraiman.