Abstract: Monte Carlo methods span a range of disciplines, drawing interest from statisticians, computer scientists, physicists, among others. In the Monte Carlo workflow, the upstream task involves designing and analyzing sampling algorithms, while the downstream task focuses on efficiently using these samples to form estimators. In this talk, I will discuss recent developments of both aspects. The first part introduces the 'Spectral Telescope' framework for analyzing Gibbs samplers, and discusses its relationship with the spectral independence technique recently developed in theoretical computer science. The second part focuses on the development of unbiased estimators through the combination of Markov Chain Monte Carlo and Multilevel Monte Carlo methods, highlighting their potential in parallel computing.