Non-destructive testing (NDT) techniques are usually used for the characterisation of defects arising in polymer composites during manufacturing or in-service use. However, each of these NDT techniques cannot always allow a full diagnosis of the material’s or component’s structural health. Thus, several techniques have to be combined in order to improve the diagnosis of the damaged state of composite structures and their evolution during the part’s life span. This opinion paper proposes a critical overview of the use and applicability of these NDT techniques for the detection and characterisation of damage to structural composite materials in view of in-service performance assessment and residual durability prognosis. It also addresses some current trends of structural health monitoring (SHM) of these materials, such as sensor–actuator embedding and NDT data fusion, and draws future perspectives on how composite SHM could evolve in the digital era, taking advantage of artificial intelligence, Internet of Things and big data to implement digital twins.