In an increasingly complex interconnected world, STEAM practices have gained attention in Higher Education (HE). The integration of Sciences, Technology, Engineering and Mathematics pedagogies with those from the Arts takes many forms with myriad intentions, processes and outcomes. Common aspirations pertain to equipping students with interdisciplinary skills required for the jobs market, increasing intellectual curiosity and collaboratively developing creative, socially equitable responses to complex global challenges. The Erasmus+ project "STEAM Innovation and Curriculum" involves seven European partners who have been pioneering STEAM approaches and methods. Using workshops, discussions, hacking processes and reflective practice, this collaborative work is the first comprehensive attempt to systematically analyse and showcase European approaches to STEAM in HE. Project outputs relate to the three project phases and include defining STEAM and a Handbook on STEAM approaches (Phase 1), the development of new STEAM methods (Phase 2) and developing evaluation frameworks (Phase 3). The Handbook collects and codifies HE STEAM approaches and provides a working definition highlighting key elements of STEAM projects and processes as well as guidance and inspiration for those looking to develop and introduce STEAM approaches in their institution. This paper focuses on Phase 1 outcomes and insights to help foster STEAM thinking and to tackle issues around joint STEM/Arts standards, and concerns in the HE arena such as modes of intellectual investigation, curriculum planning and the case for inter- and transdisciplinarity.