Abstract: The goal of the talk will be to understand what quantum expander graphs are, what they are useful for, and how they can be constructed. We will first recall the definition of classical expander graphs, and explain how quantum analogues of these objects can be defined. We will then show that, both classically and quantumly, random constructions provide with high probability examples of expander graphs. In the quantum case, such result is derived from a spectral analysis for random matrix models with a tensor product structure. If the time allows, we will present implications in terms of typical decay of correlations in 1D many-body quantum systems with local interactions.