In this tutorial, we will provide an overview of techniques and recipes for distributed learning (estimation) and testing under information constraints such as communication, local privacy, and memory constraints. Motivated by applications in machine learning and distributed computing, these questions lie at the intersection of information theory, statistics, theoretical computer science, and machine learning. We will mainly focus on minimax lower bound techniques, and cover a set of general methods to establish information-theoretic lower bounds, for both estimation and hypothesis testing questions, in both noninteractive and interactive settings.