In Markov chain Monte Carlo (MCMC), understanding the convergence properties of the underlying chain is crucial. Recently, researchers have become interested in the asymptotic behavior of convergence rates of Monte Carlo Markov chains when n or/and p grows. This gives rise to the study of convergence complexity. I will talk about some of the recent progress and challenges in the field. In particular, by analyzing Albert and Chib's algorithm for Bayesian probit model, I will explain how traditional techniques for classical convergence problems can still be powerful for convergence complexity analysis (and why they may not be powerful enough).