This thesis presents research on novel X-ray imaging methods that improve the study of specimens with small density differences, revealing their inner structure and density distribution. Exploiting the phase shift of X-rays in a material can significantly increase the image contrast compared to conventional absorption imaging. This thesis provides a practical guide to X-ray phase-contrast imaging with a strong focus on X-ray speckle-based imaging, the most recently developed phase-sensitive method. X-ray speckle-based imaging only requires a piece of abrasive paper in addition to the standard X-ray imaging setup. Its simplicity and robustness combined with the compatibility with laboratory X-ray sources, make it an ideal candidate for wide user uptake in a range of fields. An in-depth overview of the state of the art of X-ray speckle-based imaging and its latest developments is given in this thesis. It, furthermore, explores a broad range of applications, from X-ray optics characterisation, to biomedical imaging for 3D virtual histology and geological studies of volcanic rocks, demonstrating is promising potential. Moreover, the speckle-based technique is placed in the context of other phase-sensitive X-ray imaging methods to assist in the choice of a suitable method, hence serving as a guide and reference work for future users.