Lord Rees of Ludlow delivers the 2011 Romanes Lecture. Telescopes reveal the remote universe; accelerators probe the subatomic world. Thanks to such instruments, astronomers have established, in outline, how our cosmos has evolved from a still-mysterious beginning more than 13 billion years. Billions more years - and perhaps even an infinite time - lie ahead of it. But 99 percent of scientists focus neither on the very small nor the very large, but on the even greater complexities of our everyday world. Materials science, biology and the environmental sciences proceed apace, revealing remarkable insights, and opening up an ever-widening range of applications - both opportunities and threats. We live on an ever more interconnected and crowded planet, where each person is empowered by transformative technology but is making increasing demands on the world's resources. There is a widening gulf between what science enables us to do, and what it's prudent or ethical actually to do. The Earth has existed for 45 million centuries but this is the first when one species, ours, can determine the long-range planetary future. The stakes are high; optimum policies require a longer-term and less parochial perspective than normally prevails in political debate, the deployment of the best scientific advice, and engagement of a wider public. In science itself, the most dramatic conceptual advances are the least predictable. But, in scanning these intellectual horizons, we must be mindful that there may be fundamental limits to our understanding - concepts about key aspects of reality that human brains (even computer-aided) can't grasp. Lord (Martin) Rees was the President of the Royal Society from December 2005 to December 2010. He is Master of Trinity College and Professor of Cosmology and Astrophysics at the University of Cambridge. He is also Visiting Professor at Leicester University and Imperial College London. He was appointed Astronomer Royal in 1995, and was nominated to the House of Lords in 2005 as a cross-bench peer. He was appointed a member of the Order of Merit in 2007.