The dynamic academic and educational environment needs the specialists capable to work in the intensively developing open science information infrastructure, to retrieve, analyze and synthesize heterogeneous information, to process structured and unstructured research data; to document research findings, to represent them in the open access; to be engaged in social media, and to be aware of many aspects of the open science concept. The programs, forms, approaches and practice of training specialists in open science within the global initiative projects and organizations (FOSTER, Creative Commons, Digital Curation Center, COS, etc.), and individual organizations/universities (Humboldt university, Delft University of Technology) are explored. The best experience of educating open science culture online is discussed. The author also suggests the program of advanced training course “The open science culture” for scientists, post-graduatestudents, lecturers, and librarians. The program may be used in professional training and practice-oriented extended professional programs and specialized professional retraining. The study makes the background for building the system for teaching the principles and procedures of open science. The system may be implemented in academic and educational institutions.