This paper is a broad review of “quantum synchronization” — the idea that quantum systems can adjust their rhythms in ways similar to familiar classical systems. The authors collect and explain theoretical ideas, mathematical tools, example models, and experimental work that aim to extend the well‑known notion of synchronized clocks, fireflies, or pendulum clocks into the quantum domain. The review asks how synchronization shows up when systems are small, strongly quantum, or made of many interacting parts.

Rather than reporting a single new experiment or calculation, the paper organizes and summarizes the field. It explains several common approaches used to study quantum synchronization: open quantum system master equations (which follow average behavior), trajectory approaches that track individual quantum measurement records, and mean‑field or phase‑reduction methods that connect quantum models to classical nonlinear oscillators. The review also lists concrete measures researchers use to detect synchronization, including temporal correlations (how observables line up in time), quantum correlations (entanglement‑like connections), phase and frequency locking, spectral features of the system’s evolution operator (the Liouvillian), and signatures in time‑resolved emitted light.

The authors survey both few‑body examples and many‑body settings. On the few‑body side they cover quantum versions of self‑sustained oscillators, entrainment to external drives, mutual synchronization between oscillators, and phenomena such as “synchronization blockade.” For many‑body systems they discuss quantum extensions of the Kuramoto model (a standard classical synchronization model), mean‑field macroscopic synchronization, and more exotic collective behavior including time crystals. Specific physical models noted include van der Pol–type quantum oscillators, spin models, the Bose–Hubbard dimer, spin‑boson systems, lattices, and non‑reciprocal couplings.