In Chapter 8 we consider a recurrent Markov chain possessing an invariant measure which is either probabilistic in the case of positive recurrence or σ-finite in the case of null recurrence. Our main aim here is to describe the asymptotic behaviour of the invariant distribution tail for a class of Markov chains with asymptotically zero drift proportional to 1/x. We start with a result which states that a typical stationary Markov chain with asymptotically zero drift always generates a heavy-tailed invariant distribution, which is very different from the case of Markov chains with asymptotically negative drift bounded away from zero. Then we develop techniques needed for deriving precise tail asymptotics of power type.