It has long been thought that strongly correlated systems are adiabatically connected to their non-interacting counterpart. Recent developments have highlighted the fallacy of this traditional notion in a variety of settings. In this Review, we use a class of strongly correlated electron systems to illustrate the type of quantum phases and fluctuations that are created by strong correlations. Examples include quantum critical states that violate the Fermi liquid paradigm, unconventional superconductivity that goes beyond the Bardeen–Cooper–Schrieffer framework, and topological semimetals induced by the Kondo interaction. We assess the prospects for designing other exotic phases of matter by using alternative degrees of freedom or alternative interactions, and discuss the potential of these correlated states for quantum technology.