Slow-light fiber Bragg gratings (FBGs) belong to a class of gratings designed to exhibit one or more narrow resonances in their reflection and transmission spectra, produced either by introducing a pi phase shift near the middle of the grating, or by increasing the index modulation so that the grating behaves like a Fabry-Perot interferometer. These resonances can have very narrow linewidths (<50 fm), resulting in low group velocities and high Q factors. Slow-light gratings are finding a growing number of applications in many areas of photonics, including nonlinear optics, optical switching, optical delay lines, and sensing. This paper reviews the principle of these gratings, in particular the more recent slow-light gratings relying on a strong index modulation. It discusses in particular the requirements for achieving large group delays and high sensitivities in sensors, and the fabrication and annealing techniques used to meet these requirements (high index modulation, low loss, index-profile apodization, and optimized length). Several applications are presented, including record-breaking FBGs that exhibit a group delay of 42 ns and Q-factor of similar to 30 million over a 12.5 mm length, robust acoustic sensors with pressure resolution of similar to 50 mu Pa (root Hz)(-1) in the few-kHz, and a strain sensor capable of resolving as little as 30 femtostrain (root Hz)(-1).