Future wireless services will focus on improving the quality of life by enabling various applications, such as extended reality, brain-computer interaction, and healthcare. These applications will have diverse performance requirements (e.g., user-defined quality of experience metrics, latency, and reliability) which will be challenging to be fulfilled by existing wireless systems. To meet the diverse requirements of the emerging applications, the concept of digital twins has been recently proposed. A digital twin uses a virtual representation along with security-related technologies (e.g., blockchain), communication technologies (e.g., 6G), computing technologies (e.g., edge computing), and machine learning, so as to enable the smart applications. In this tutorial, we present a comprehensive overview on digital twins for wireless systems. First, we present the fundamental concepts (i.e., design aspects, high-level architecture, and frameworks) of digital twins for wireless systems. Second, a comprehensive taxonomy is devised for two aspects, namely, twins for wireless and wireless for twins. For the twins for wireless aspect, we consider issues related to design of twin objects, physical devices, and interface, as well as prototyping, deployment trends, incentive mechanism, isolation of twins, and decoupling. For the wireless for twins aspect, we consider issues related to accessing twin objects, security and privacy, and air interface design are considered. Finally, open research challenges and opportunities are discussed. IEEE