In the last three decades, the development and steady improvement of various optical technologies at the near-infrared region of the electromagnetic spectrum has inspired a large number of scientists around the world to design and develop functional near-infrared spectroscopy (fNIRS) systems for various medical applications. This has been driven further by the availability of new sources and detectors that support very compact and wearable system designs. In this article, we review fNIRS systems from the instrumentation point of view, discussing the associated challenges and state-of-the-art approaches. In the beginning, the fundamentals of fNIRS systems as well as light-tissue interaction at NIR are briefly introduced. After that, we present the basics of NIR systems instrumentation. Next, the recent development of continuous-wave, frequency-domain, and time-domain fNIRS systems are discussed. Finally, we provide a summary of these three modalities and an outlook into the future of fNIRS technology.