After many years of rigid conventional procedures of production, industrial manufacturing is going through a process of change toward flexible and intelligent manufacturing, the so-called Industry 4.0. In this paper, human–robot collaboration has an important role in smart factories since it contributes to the achievement of higher productivity and greater efficiency. However, this evolution means breaking with the established safety procedures as the separation of workspaces between robot and human is removed. These changes are reflected in safety standards related to industrial robotics since the last decade, and have led to the development of a wide field of research focusing on the prevention of human–robot impacts and/or the minimization of related risks or their consequences. This paper presents a review of the main safety systems that have been proposed and applied in industrial robotic environments that contribute to the achievement of safe collaborative human–robot work. Additionally, a review is provided of the current regulations along with new concepts that have been introduced in them. The discussion presented in this paper includes multi-disciplinary approaches, such as techniques for estimation and the evaluation of injuries in human–robot collisions, mechanical and software devices designed to minimize the consequences of human–robot impact, impact detection systems, and strategies to prevent collisions or minimize their consequences when they occur.