Abrasive flow finishing (AFF) is a non-traditional advanced fine finishing process using an abrasive-laden viscoelastic polymer for finishing, polishing, deburring, radiusing of the component having a complex geometrical shape with inaccessible areas and difficult to machine materials. AFF is a consistent, accurate and repeatable process compared to another finishing process. It was initially developed in 1960 by Extrude Hone Corporation, USA for finishing valves of radar-guided vehicles. It uses a flexible abrasive laden viscoelastic finishing medium, which makes it most popular with nanofinishing process for simple as well as complex geometrical components. Continual research in many areas of AFF is in underway to explore its applications for different components, finishing medium developments and applications, process modeling and online control, automation with robotic interface for mass finishing operation, development of new mechanisms and types for fourth-generation AFF. This article reports on advances in process modeling and optimization, alternative finishing medium and rheological characterization, hybrid, derived and hybrid-derived variants and applications of AFF for finishing advanced materials and complicated geometry. This article will be beneficial for researchers, academicians, and industrialists who are working in the field of precision finishing of complex geometrical mechanical components and willing to implement AFF.