Out of equilibrium, the lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter, nonequilibrium systems, networks of neurons, social groups with conformist and contrarian members, and directional interface growth phenomena. While wave propagation in non-reciprocal media has recently been under intense study, less is known about the consequences of non-reciprocity on the collective behavior of many-body systems. Here, we show that non-reciprocity leads to time-dependent phases where spontaneously broken continuous symmetries are dynamically restored. The resulting phase transitions are controlled by spectral singularities called exceptional points. We describe the emergence of these phases using insights from bifurcation theory and non-Hermitian quantum mechanics. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point enforced pattern-formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in non-reciprocal matter.