Nowadays, smart composite materials embed miniaturized sensors for structural health monitoring (SHM) in order to mitigate the risk of failure due to an overload or to unwanted inhomogeneity resulting from the fabrication process. Optical fiber sensors, and more particularly fiber Bragg grating (FBG) sensors, outperform traditional sensor technologies, as they are lightweight, small in size and offer convenient multiplexing capabilities with remote operation. They have thus been extensively associated to composite materials to study their behavior for further SHM purposes. This paper reviews the main challenges arising from the use of FBGs in composite materials. The focus will be made on issues related to temperature-strain discrimination, demodulation of the amplitude spectrum during and after the curing process as well as connection between the embedded optical fibers and the surroundings. The main strategies developed in each of these three topics will be summarized and compared, demonstrating the large progress that has been made in this field in the past few years.