Composite materials/structures are advancing in product efficiency, cost-effectiveness and the development of superior specific properties. There are increasing demands in their applications to load-carrying structures in aerospace, wind turbines, transportation, medical equipment and so on. Thus, robust and reliable non-destructive testing of composites is essential to reduce safety concerns and maintenance costs. There have been various non-destructive testing methods built upon different principles for quality assurance during the whole lifecycle of a composite product. This article reviews the most established non-destructive testing techniques for detection and evaluation of defects/damage evolution in composites. These include acoustic emission, ultrasonic testing, infrared thermography, terahertz testing, shearography, digital image correlation, as well as X-ray and neutron imaging. For each non-destructive testing technique, we cover a brief historical background, principles, standard practices, equipment and facilities used for composite research. We also compare and discuss their benefits and limitations and further summarise their capabilities and applications to composite structures. Each non-destructive testing technique has its own potential and rarely achieves a full-scale diagnosis of structural integrity. Future development of non-destructive testing techniques for composites will be directed towards intelligent and automated inspection systems with high accuracy and efficient data processing capabilities.