The ability to process optical signals without passing into the electrical domain has always attracted the attention of the research community. Processing photons by photons unfolds new scenarios, in principle allowing for unseen signal processing and computing capabilities. Optical computation can be seen as a large scientific field in which researchers operate, trying to find solutions to their specific needs by different approaches; although the challenges can be substantially different, they are typically addressed using knowledge and technological platforms that are shared across the whole field. This significant know-how can also benefit other scientific communities, providing lateral solutions to their problems, as well as leading to novel applications. The aim of this Roadmap is to provide a broad view of the state-of-the-art in this lively scientific research field and to discuss the advances required to tackle emerging challenges, thanks to contributions authored by experts affiliated to both academic institutions and high-tech industries. The Roadmap is organized so as to put side by side contributions on different aspects of optical processing, aiming to enhance the cross-contamination of ideas between scientists working in three different fields of photonics: optical gates and logical units, high bit-rate signal processing and optical quantum computing. The ultimate intent of this paper is to provide guidance for young scientists as well as providing research-funding institutions and stake holders with a comprehensive overview of perspectives and opportunities offered by this research field.