The aim of this review is to present the state of the art of neurophotonics, a recently founded discipline lying at the interface between optics and neuroscience. While neurophotonics also includes invasive techniques for animal studies, in this review we focus only on the non-invasive methods that use near infrared light to probe functional activity in the brain, namely the fast optical signal, diffuse correlation spectroscopy, and functional near infrared spectroscopy methods. We also present an overview of the physical principles of light propagation in biological tissues, and of the main physiological sources of signal. Finally, we discuss the open issues in models, instrumentation, data analysis and clinical approaches.