We investigate the degree to which impact in science and technology is associated with surprising breakthroughs, and how those breakthroughs arise. Identifying breakthroughs across science and technology requires models that distinguish surprising from expected advances at scale. Drawing on tens of millions of research papers and patents across the life sciences, physical sciences and patented inventions, and using a hypergraph model that predicts realized combinations of research contents (article keywords) and contexts (cited journals), here we show that surprise in terms of unexpected combinations of contents and contexts predicts outsized impact (within the top 10% of citations). These surprising advances emerge across, rather than within researchers or teams—most commonly when scientists from one field publish problem-solving results to an audience from a distant field. Our approach characterizes the frontier of science and technology as a complex hypergraph drawn from high-dimensional embeddings of research contents and contexts, and offers a measure of path-breaking surprise in science and technology.