The accretion-induced collapse (AIC) scenario was proposed 40 years ago as an evolutionary end state of oxygen-neon white dwarfs (ONe WDs), linking them to the formation of neutron star (NS) systems. However, there has been no direct detection of any AIC event so far, even though there exists a lot of indirect observational evidence. Meanwhile, the evolutionary pathways resulting in NS formation through AIC are still not thoroughly investigated. In this article, we review recent studies on the two classic progenitor models of AIC events, i.e., the single-degenerate model (including the ONe WD+MS/RG/He star channels and the CO WD+He star channel) and the double-degenerate model (including the double CO WD channel, the double ONe WD channel and the ONe WD+CO WD channel). Recent progress on these progenitor models is reviewed, including the evolutionary scenarios leading to AIC events, the initial parameter space for producing AIC events and the related objects (e.g., the pre-AIC systems and the post-AIC systems). For the single-degenerate model, the pre-AIC systems (i.e., the progenitor systems of AIC events) could potentially be identified as supersoft X-ray sources, symbiotics and cataclysmic variables (such as classical novae, recurrent novae, Ne novae and He novae) in the observations, whereas the post-AIC systems (i.e., NS systems) could potentially be identified as low-/intermediate-mass X-ray binaries, and the resulting low-/intermediate-mass binary pulsars, most notably millisecond pulsars. For the double-degenerate model, the pre-AIC systems are close double WDs with short orbital periods, whereas the post-AIC systems are single isolated NSs that may correspond to a specific kind of NS with peculiar properties. We also review the predicted rates of AIC events, the mass distribution of NSs produced via AIC and the gravitational wave (GW) signals from double WDs that are potential GW sources in the Galaxy in the context of future space-based GW detectors, such as LISA, TianQin, Taiji, etc. Recent theoretical and observational constraints on the detection of AIC events are summarized. In order to confirm the existence of the AIC process, and resolve this long-term issue presented by current stellar evolution theories, more numerical simulations and observational identifications are required.