Computing the stationary distributions of a continuous-time Markov chain
involves solving a set of linear equations. In most cases of interest, the
number of equations is infinite or too large, and cannot be solved analytically
or numerically. Several approximation schemes overcome this issue by truncating
the state space to a manageable size. In this review, we first give a
comprehensive theoretical account of the stationary distributions and their
relation to the long-term behaviour of the Markov chain, which is readily
accessible to non-experts and free of irreducibility assumptions made in
standard texts. We then review truncation-based approximation schemes paying
particular attention to their convergence and to the errors they introduce, and
we illustrate their performance with an example of a stochastic reaction
network of relevance in biology and chemistry. We conclude by elaborating on
computational trade-offs associated with error control and some open questions.