The interest of studying the formation and evolution of compact objects in the accretion disks of active galactic nuclei (AGNs) dates back to the 21st century. The observational evidence of early metal enrichment in AGNs as well as the variability of AGNs motivated people then to theoretically investigates the formation and motion of compact objects in the accretion disks. More recently, the rapid progress in the observation of gravitational waves (GWs) revived the interest in the above studies. In particular, the high masses of the binary black holes (BHs) discovered by the Laser Interferometer Gravitational-wave Observatory (LIGO) and the Virgo detectors challenge the current standard model of binary BH (BBH) formation, and hence require new insights in the origin of these binaries. The accretion disk of AGNs is a favorable location for such heavy binaries to form because of the possibility of gas accretion and pairing with other BHs. Previous theoretical work on the formation and evolution of stellar-mass objects (e.g., stars, stellar-mass BHs, etc.) in AGN disks found that migration is one of the most important ingredients in the model that determines the characters of the LIGO/Virgo BBHs. Here, migration stands for the gradual change of the orbital radius of the small object due to its interaction with the surrounding gas or other stellar objects. In this review, we summarize the theoretical and observational progress related to the migration of compact objects in the accretion disks of AGNs, and point to the important problems which need to be addressed in the future. We firstly review three major types of migration which have been investigated in the literature. (1) For the compact objects whose orbits are inclined and not embedded in the disk, every time they hit the accretion disk, the orbital angular momenta are lost due to the collision. (2) Compact objects can excite density waves in the disk, which can transport angular momenta away from the compact objects via the backreaction on the objects. (3) In the case where the orbital velocity of the gas in the disk is different from the Keplerian velocity, the differential rotation between the gas and the embedded stellar object results in a hydro-dynamical friction which can also cause migration. In addition, we also briefly describe some new migration mechanisms considered recently such as thermal torque and migration driven by third body. After reviewing the theory of migration, we further summarize the possible observational signatures related to migration. They include: (1) Metal enrichment in AGNs, induced by supernova explosion in the final evolutionary stage of an embedded star; (2) the formation of BBHs by multiple BH encounters following differential migration, and the subsequent GW radiation; (3) formation of extreme-mass-ratio or intermediate-mass-ratio inspirals when stellar-mass BHs or IMBHs finally migrate to the vicinity of the central SMBH, as well as the subsequent dynamical interaction and GW radiation in the milli-Hertz GW band. Finally, we conclude this review by highlighting the important theoretical problems and the possible directions worth exploring in the future.
