This chapter focuses on gradient-based optimization methods for solving learning problems in federated learning systems. These methods update model parameters using local information about the gradient of the objective function. The chapter begins with a review of the basic gradient step and explains how to choose the learning rate and stopping criteria. It then discusses how to handle noisy or incomplete gradient information, which is common in distributed settings. To support constrained optimization tasks, the projected gradient method is introduced. The chapter also extends gradient methods to non-parametric models by using test datasets and replacing gradients with more general update rules. Finally, gradient-based methods are interpreted as fixed-point iterations, offering a unified view of many federated learning algorithms. Throughout, the emphasis is on practical implementation and theoretical guarantees. These techniques provide the foundation for efficient, scalable, and flexible optimization in federated learning, especially when working with large models or under limited communication and data-sharing constraints.