This thesis is focused on an advanced imaging method X-ray computed nanotomography (CT). This non-destructive technique is used for the research of various biomaterials in tissue engineering and material science in general (scaffolds, polymers, ceramics, composites, etc.). Visualisation and quantification in 3D are advantageous in multidisciplinary approach usually applied in these fields. The objective of the thesis is divided into two topics. The first topic is about optimisation of the measurement procedure for various soft materials by CT with the laboratory X-ray sources. Mostly, the phase contrast propagation-based CT imaging (PBI) is involved here. This work theoretically describes the PBI and demonstrates this phenomenon via several sets of measurements. The necessary post-processing of PBI data is implemented and evaluated based on data quality enhancement. The second topic shows specific applications of CT in material engineering. Several studies with different CT devices show a few examples of possible applications and image processing options. Examples of correlation of CT with other complementary techniques show how CT can be applied in a multidisciplinary approach to solving complex scientific problems.