Since the seminal work of Roentgen, X-ray imaging mainly uses the same physical phenomenon: the absorption of light by matter. Thanks to third-generation synchrotrons that provide a high flux of quasi-coherent X-rays, we have seen in recent years new imaging concepts such as phase contrast or dark-field imaging that were later adapted to conventional X-ray sources. These innovative imaging techniques are particularly suitable for visualizing soft matter, such as biological tissues. After a brief introduction to the physical foundations of these two techniques, we present the different experimental set-ups that are now available to produce such contrasts: propagation, analyzer-based, grating interferometry and non-interferometric methods, such as coded aperture and modulation techniques. We present a comprehensive review of their principles; associated data processing; and finally, their requirements for their transfer outside of synchrotrons. In conclusion, gratings interferometry, coded aperture and modulation techniques seem to be the best candidates for the widespread use of phase contrast and dark-field imaging on low-cost X-ray sources.