X‐ray computed tomography is an important tool for non‐destructively evaluating the 3‐D microstructure of modern materials. To resolve material structures in the micrometer range and below, high brilliance synchrotron radiation has to be used. The Federal Institute for Materials Research and Testing (BAM) has built up an imaging setup for micro‐tomography and ‐radiography (BAMline) at the Berliner storage ring for synchrotron radiation (BESSY). In computed tomography, the contrast at interfaces within heterogeneous materials can be strongly amplified by effects related to X‐ray refraction. Such effects are especially useful for materials of low absorption or mixed phases showing similar X‐ray absorption properties that produce low contrast. The technique is based on ultra‐small‐angle scattering by microstructural elements causing phase‐related effects, such as refraction and total reflection. The extraordinary contrast of inner surfaces is far beyond absorption effects. Crack orientation and fibre/matrix debonding in plastics, polymers, ceramics and metal‐matrix‐composites after cyclic loading and hydro‐thermal aging can be visualized. In most cases, the investigated inner surface and interface structures correlate to mechanical properties. The technique is an alternative to other attempts on raising the spatial resolution of CT machines.