Fiber-reinforced polymers (FRPs) are low-density, high-performance composite materials, which find important applications in the automotive, aerospace, and energy industry, to only cite a few. With the increasing concerns about sustainability and environment risks, the problem of the recycling of such complex composite systems has been emerging in politics, industry, and academia. The issue is exacerbated by the increased use of FRPs in the automotive industry and by the expected decommissioning of airplanes and wind turbines amounting to thousands of metric tons of composite materials. Currently, the recycling of FRPs downcycles the entire composite to some form of reinforcement material (typically for cements) or degrades the polymer matrix to recover the fibers. Following the principles of sustainability, the reuse and recycling of the whole composite—fiber and polymer—should be promoted. In this review paper, we report on recent research works that achieve the recycling of both the fiber and matrix phase of FRP composites, with the polymer being either directly recovered or converted to value-added monomers and oligomers.