The revival of interest in Markov chains is based in part on their recent applicability in solving real world problems and in part on their ability to resolve issues in theoretical computer science. This paper presents three examples which are used to illustrate both parts: a Markov chain algorithm for estimating the tails of the bootstrap also illustrates the Jerrum-Sinclair theory of approximate counting. The Geyer-Thompson work on Monte-Carlo evaluation of maximum likelihood is compared with work on evaluation of the partition function. Finally, work of Diaconis-Sturmfels on conditional inference is complemented by the work of theoretical computer scientists on approximate computation of the volume of convex polyhedra.