Intense ultrashort laser fields can prepare quantum coherence and multi-body entanglement in atoms and molecules on a sub-cycle timescale by combining broadband excitation with resonant coupling. Unlike conventional quantum optics, where coherence is built through the resonant interaction of weak narrowband light with atoms, here the coherence is created in an open, nonequilibrium system that undergoes ionization and radiates while the driven process is still under way. This review surveys the emerging field of strong-field quantum optics, covering the creation, diagnosis and application of such coherence. Five elementary processes are analyzed as sources of strong-field coherence: ionization and excitation that leave the ion in a superposition of electronic states, vibrational and rotational wave-packet formation in molecules, nonadiabatic coupling that redistributes coherence among electronic and nuclear degrees of freedom, entanglement between the photoelectron and the parent ion, and multi-electron coherence that initiates charge migration. These processes give rise to novel coherent emissions, including N2+ air lasing, ultraviolet supercontinuum radiation and free induction decay. The gain mechanism of N2+ air lasing remains controversial, and competing scenarios including population inversion, lasing without inversion, superradiance and post-pulse free induction decay are compared. We present an ionization-coupling theoretical framework that treats strong-field ionization and subsequent laser-ion coupling on the same footing, and discuss the challenges of extending it to many-electron, multi-degree-of-freedom open systems and to collective emission. On the experimental side, multidimensional spectroscopy, attosecond transient-absorption spectroscopy and photoelectron momentum spectroscopy are reviewed as state-resolved probes of coherence dynamics. Finally, the unique properties of air lasing and its potential applications in remote sensing, trace-gas detection and combustion diagnostics are discussed. Realizing air lasing as a practical radiation source with controllable temporal, spectral and spatial properties requires improving the output brightness and suppressing the supercontinuum background generated during filamentation.