The goal of this course is to give an introduction of trace inequalities and quantum entropies, with a special focus on inequalities and convexity/concavity results related to quantum entropies that have played important roles in the history. They have strong applications in many areas like quantum information and mathematical physics. We will not only review some classical results such as Lieb’s concavity theorem and Strong subadditivity of quantum entropy, but also discuss some recent progress such as data processing inequalities for quantum relative entropies.