This tutorial introduces the theoretical and experimental aspects of quantum optimal control (QOC), focusing on methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. Based on the Pontryagin Maximum Principle (PMP), the paper presents the basic elements of QOC in a physicist-friendly manner, drawing an analogy with classical Lagrangian and Hamiltonian mechanics. It emphasizes various numerical algorithms for solving quantum optimal control problems, providing codes based on the shooting method and gradient-based algorithms. The connection between optimal processes and the quantum speed limit is discussed in two-level quantum systems, while the experimental implementation of optimal control protocols is described for two-level and many-level systems, including Bose-Einstein Condensates (BEC) in one-dimensional optical lattices. The paper also highlights the challenges in transferring theoretical results to experimental implementations, such as the need for precise models and consideration of experimental constraints. It discusses the application of QOC in various quantum technologies, including quantum computing and sensing, and its growing importance in the field. The paper is organized into sections covering optimal control theory, numerical methods, and applications to quantum systems, with a focus on the PMP and its role in solving optimal control problems. It also addresses the connection between QOC and the quantum speed limit, as well as the use of numerical methods in quantum systems. The paper concludes with a discussion of the future directions of QOC and its potential applications in quantum technologies.This tutorial introduces the theoretical and experimental aspects of quantum optimal control (QOC), focusing on methods for designing time-varying electromagnetic fields to perform operations in quantum technologies. Based on the Pontryagin Maximum Principle (PMP), the paper presents the basic elements of QOC in a physicist-friendly manner, drawing an analogy with classical Lagrangian and Hamiltonian mechanics. It emphasizes various numerical algorithms for solving quantum optimal control problems, providing codes based on the shooting method and gradient-based algorithms. The connection between optimal processes and the quantum speed limit is discussed in two-level quantum systems, while the experimental implementation of optimal control protocols is described for two-level and many-level systems, including Bose-Einstein Condensates (BEC) in one-dimensional optical lattices. The paper also highlights the challenges in transferring theoretical results to experimental implementations, such as the need for precise models and consideration of experimental constraints. It discusses the application of QOC in various quantum technologies, including quantum computing and sensing, and its growing importance in the field. The paper is organized into sections covering optimal control theory, numerical methods, and applications to quantum systems, with a focus on the PMP and its role in solving optimal control problems. It also addresses the connection between QOC and the quantum speed limit, as well as the use of numerical methods in quantum systems. The paper concludes with a discussion of the future directions of QOC and its potential applications in quantum technologies.