Bilevel optimization has widespread applications in machine learning and data science, e.g., hyperparameter optimization, neural network architecture search, meta learning, etc. To facilitate those machine learning models to federated learning, federated bilevel optimization has been actively studied recently. To deepen the understanding of federated bilevel optimization and advance its development, this chapter discusses the unique challenges, state-of-the-art (SOTA) algorithms, and recent advances in federated bilevel optimization. Especially, this chapter demonstrates how SOTA algorithms approximate hypergradient under different settings to make federated bilevel optimization feasible.