In this talk I will give a brief introduction to the convexity and concavity of trace functionals involving trace and matrices. They play an important role in quantum information theory. Since Lieb's celebrated work in 1973 resolving the conjecture of Wigner-Yanase-Dyson, this topic has seen great progress. As an example, I will introduce a conjecture in quantum information theory by Audenaert-Datta and a stronger conjecture by Carlen-Frank-Lieb in recent years, explain the connection with convexity/concavity of trace functionals, and show how to solve them in a simple way. The key ingredients are some variational formulas coming from basic properties of Schatten norms. This is based on arXiv:1811.01205.