The theory of second-order phase transitions is one of the foundations of modern statistical mechanics and condensed-matter theory. A central concept is the observable order parameter, whose nonzero average value characterizes one or more phases. At large distances and long times, fluctuations of the order parameter(s) are described by a continuum field theory, and these dominate the physics near such phase transitions. We show that near second-order quantum phase transitions, subtle quantum interference effects can invalidate this paradigm, and we present a theory of quantum critical points in a variety of experimentally relevant two-dimensional antiferromagnets. The critical points separate phases characterized by conventional “confining” order parameters. Nevertheless, the critical theory contains an emergent gauge field and “deconfined” degrees of freedom associated with fractionalization of the order parameters. We propose that this paradigm for quantum criticality may be the key to resolving a number of experimental puzzles in correlated electron systems and offer a new perspective on the properties of complex materials.