The third-generation synchrotron radiation sources are widely used in physics, chemistry, material science, etc. due to their light beams with high brilliance and low emittance. In order to efficiently utilize such light beams for scientific research, reflective mirrors with excellent figure quality are required. The reflective mirrors on the beamlines of synchrotron radiation sources consist of fixed polished shape mirrors and bendable mirrors. Bendable mirrors have been attracting the attention of the synchrotron radiation community because their curvatures can be varied to realize different focusing properties. Classical bendable mirrors are realized by applying mechanical moment at the ends of the mirror substrates. In this paper, we introduce a new concept of bendable mirrors, X-ray adaptive mirrors which are based on the adaptive optics technology and the properties of piezoelectric bimorph systems. X-ray adaptive mirrors exhibit many advantages over the classical bendable mirrors, such as mechanics-free, figure local corrections, and good focusing properties. The piezoelectric bimorph mirrors have been used in astronomy to correct the wavefront distortions introduced by atmospheric turbulence in real time. The piezoelectric bimorph mirror was first introduced into the field of synchrotron radiation by European Synchrotron Radiation Facility (ESRF) in the 1990s for making an X-ray reflective mirror. Compared with astronomy community, synchrotron radiation community is not interested in high-speed wavefront correction, but looking for the ultimate precision of the surface shape of piezoelectric bimorph mirror. In the second part of this paper, the usual structure and working principle are briefly described. Piezoelectric bimorph mirrors are laminated structures consisting of two strips of an active material such as zirconate lead titanate (PZT) and two faceplates of a reflecting material such as silicon. A discrete or continuous control electrode is located between the interfaces of PZT-PZT, while two continuous ground electrodes are located between the interfaces of Si-PZT. The PZTs that are polarized normally to their surface, any voltage applied across the bimorph results in a different change of the lateral dimensions of two PZTs, thereby leading to a bending of the whole structure. The relationship between the curvature of the bending mirror and voltage is given. In the third part of this paper, the technical issues as well as the design concepts are discussed in detail. Several Si-PZT-PZT-Si bimorph mirrors are first fabricated and tested by ESRF. The dimensions of each of them are 150 mm in length, 4045 mm in width, and 1518 mm in thickness. PZT is selected as an active material because of its high coupling factor, high piezoelectric coefficient, and high Curie temperature. The faceplates need to be easy to polish such as silicon and silica. Owing to the symmetrical layered structure Si-PZT-PZT-Si, the mirror is less sensitive to temperature variations from the process of bonding and polishing. The bimorph mirrors are confirmed to be promising by experimental tests. As the state-of-art polishing technique, elastic emission machining (EEM) becomes available commercially, and diamond light source brings EEM into the bimorph mirror to achieve a novel adaptive X-ray mirror coupling adaptive zonal control with a super-smooth surface. This super-polished adaptive mirror becomes the first optics with a bendable ellipse with sub-nanometer figure error. Spring-8 fabricates an adaptive mirror with different structures, and two strips of PZTs are glued to the side faces of the mirror. This mirror shows a diffraction-limited performance. Finally, the wavefront measuring methods and control algorithm are introduced. Wavefront measuring devices used in the metrology cleanroom include long trace profiler, nanometer optics component measuring machine, and interferometer. At-wavelength measuring methods used on the beamline include pencil-beam method, phase retrieval method, X-ray speckle tracking technique, and Hartmann test. The wavefront control algorithm is aimed at obtaining the voltages applied according to the inverse of the interaction matrix.