The article considers prerequisites and methodological presumptions inherent in the notion of quality culture, and argues for the necessity to more systematically include quality teaching maintenance in this notion. Presented are the goals, methods, and basic results of the first project «Supporting Quality Teaching in Higher Education» carried out by the «Institutional Governance in Higher Education» OECR program. On the basis of the colleges participating in the program, the article considers priority directions in the ensuring of the quality teaching, as well as the background and context of the development of various moves to ensure the quality teaching, and describes its participants and mechanisms. The outcomes of the participation of HSE in the first stage of the project are laid out.