In this article we describe a new coupling technique which is useful in a variety of perfect sampling algorithms. A multishift coupler generates a random function f() so that for each real x, f(x)-x is governed by the same fixed probability distribution, such as a normal distribution. We develop the class of layered multishift couplers, which are simple and have several useful properties. For the standard normal distribution, for instance, the layered multishift coupler generates an f() which (surprisingly) maps an interval of length L to fewer than 2+L/2.35 points --- useful in applications which perform computations on each such image point. The layered multishift coupler improves and simplifies algorithms for generating perfectly random samples from several distributions, including the autogamma distribution, posterior distributions for Bayesian inference, and the steady state distribution for certain storage systems. We also use the layered multishift coupler to develop a Markov-chain based perfect sampling algorithm for the autonormal distribution. At the request of the organizers, we begin by giving a primer on CFTP (coupling from the past); CFTP and Fill's algorithm are the two predominant techniques for generating perfectly random samples using coupled Markov chains.