  Motivated by recent studies of large Mallows$(q)$ permutations, we propose a
class of random permutations of $\mathbb{N}_{+}$ and of $\mathbb{Z}$, called
regenerative permutations. Many previous results of the limiting Mallows$(q)$
permutations are recovered and extended. Three special examples: blocked
permutations, p-shifted permutations and p-biased permutations are studied.
