Rich countries have developed a historically unprecedented capability to manage conventional risks - fire, floods, earthquakes etc., but also car accidents, many workplace risks, and more. It is based on two institutions - insurance markets and public risk governance - supported by a powerful theory: the expected utility approach to risk. Expected utility refines the utilitarian paradigm of rational action by combining the concept of utility functions with the concept of probability distributions, using subjective probabilities where required. One might think that future progress in risk research will consist mainly in refining this approach and spreading it to emerging and less developed countries. However, greater progress is necessary and possible. It is necessary because the global economy and technostructure we live in have generated new systemic risks - including financial crises, pandemics, climate change, nuclear war. These risks exceed the coping capacity of conventional risk management and call for new forms of integrated risk governance. Greater progress is possible because recent research has developed ways to address the basic difficulties of expected utility without loosing its valuable insights. They involve three major advances. First, to introduce a risk function that generalizes expected utility so as to overcome well-known difficulties like the Allais paradox. Second, to embed expected utility in a framework of iterated network games so as to take into account the social learning processes that are essential for real world risk governance. And third, to accommodate the logic of complementary descriptions called for by the new systemic risks of the 21st century. The coming breakthrough in risk research may best be achieved by bringing these advances to bear on practical efforts aiming at integrated risk governance.