With the development of communications technologies, the remarkable progress of IoT and the growth of Industry 4.0 in recent years, research in cybersecurity is of increasing interest in industry and economic field. Moreover, quantum computing threatens to break traditional cryptographic systems and traditional cybersecurity schemes. Therefore, it is essential to deploy new solutions in order to overcome this challenge. In this paper we address some of the most relevant aspects of the field of cybersecurity in the context of the development of quantum technologies, both in terms of challenges to be overcome and new opportunities. In this sense, the article is divided into several sections that address the different aspects. Each of the sections consists of two parts. The first part consists of a brief bibliographical review. The second part introduces a framework and a roadmap to advance in the investigation of each of them in the future.