Neural signal recording is critical in modern day neuroscience research and emerging neural prosthesis programs. Neural recording requires the use of precise, low-noise amplifier systems to acquire and condition the weak neural signals that are transduced through electrode interfaces. Neural amplifiers and amplifier-based systems are available commercially or can be designed in-house and fabricated using integrated circuit (IC) technologies, resulting in very large-scale integration or application-specific integrated circuit solutions. IC-based neural amplifiers are now used to acquire untethered/portable neural recordings, as they meet the requirements of a miniaturized form factor, light weight and low power consumption. Furthermore, such miniaturized and low-power IC neural amplifiers are now being used in emerging implantable neural prosthesis technologies. This review focuses on neural amplifier-based devices and is presented in two interrelated parts. First, neural signal recording is reviewed, and practical challenges are highlighted. Current amplifier designs with increased functionality and performance and without penalties in chip size and power are featured. Second, applications of IC-based neural amplifiers in basic science experiments (e.g., cortical studies using animal models), neural prostheses (e.g., brain/nerve machine interfaces) and treatment of neuronal diseases (e.g., DBS for treatment of epilepsy) are highlighted. The review concludes with future outlooks of this technology and important challenges with regard to neural signal amplification.