  This review paper provides an introduction of Markov chains and their
convergence rates which is an important and interesting mathematical topic
which also has important applications for very widely used Markov chain Monte
Carlo (MCMC) algorithm. We first discuss eigenvalue analysis for Markov chains
on finite state spaces. Then, using the coupling construction, we prove two
quantitative bounds based on minorization condition and drift conditions, and
provide descriptive and intuitive examples to showcase how these theorems can
be implemented in practice. This paper is meant to provide a general overview
of the subject and spark interest in new Markov chain research areas.
