It is generally believed that the electron-capture reactions happen when the oxygen-neon (ONe) cores grow in masses close to the Chandrasekhar limit, leading to the formation of neutron stars (NSs) via electron-capture supernovae (EC-SNe). EC-SNe are predicted to be the most likely short-lived and faint optical transients, and a small ejecta mass is expected during the collapse. This kind of SNe provide a distinct channel for producing isolated NSs and NS systems, especially for the formation of X-ray binaries and double NSs. Although EC-SNe were proposed ∼45 yr ago, there are still some uncertainties for the origin of EC-SNe and their productions. In this article, we review recent studies on the two classic progenitor channels of EC-SNe, i.e., the single star channel and the binary channel. In the single star channel, EC-SNe can happen in super asymptotic giant branch stars or He stars, whereas in the binary channel EC-SNe can occur in He stars in binaries (involving He star+MS systems and NS+He star systems) or accretion-induced collapse in white dwarf binaries (involving the single-degenerate scenario and the double-degenerate scenario). Recent progress on these two progenitor channels is discussed, including the initial parameter range for EC-SNe, the evolutionary paths to EC-SNe, related objects and some observational constraints, etc. We also make some discussions on the possible candidates for EC-SNe in this article, and the impacts of EC-SNe on some research fields, e.g., the properties of NSs, double NS population and chemical products, etc. It is noting that EC-SNe show some similar properties with ultra-stripped SNe, e.g., low ejecta masses and small kicks. Accordingly, we also discuss the difference between these two types of SNe in this article. Research on EC-SNe is at a pivotal stage, with key theoretical uncertainties and observational challenges requiring integrated modeling and multi-wavelength observations for robust identification.