Learn about the groundbreaking developments in Markov-chain algorithms through this comprehensive lecture delivered at the Galileo Galilei Institute. Explore the fundamental concepts and recent advancements that constitute what's being called the second Markov-chain revolution. Delve into cutting-edge mathematical and computational techniques that have transformed our understanding of these powerful algorithmic tools. Over 82 minutes, gain insights into how these revolutionary changes are reshaping the landscape of statistical physics, machine learning, and computational mathematics.