Timely detection of barely visible impact damage (BVID) in composite structures is paramount to prevent catastrophic failure. Traditionally, BVID detection has been carried out in non-destructive (NDT) inspections. Nevertheless, damage detection in some applications may not be enough. For example, estimation of damage location, extent, failure mode, or prognosis of remaining useful life may be desirable. The information obtained on damage can be potentially enhanced by employing more universal or specialized signal processing algorithms. This review provides a detailed description of the workings of several NDT techniques, such as ultrasonics, X-ray tomography, thermography, optical shearography, and optical computed tomography, concerning impact damage detection. These NDT techniques are then compared in terms of their performance, such as sensitivity and resolution, speed, complexity, and cost. A particular emphasis is put on signal processing algorithms that are established to enrich damage characterization, along with their advantages and limitations. Cases of combining several NDT techniques in enhancing BVID detection and the merits of such approaches are explored. Discussion of possible ways of increasing safety of composite structures, for example, implementation of artificial intelligence for enhanced inspections, manufacturing of advanced BVID-resistant composites and integration of robotized systems for increased data acquisition is provided.