Because the phase contains more information about the field in contrast to the amplitude, phase measurement has always been a hot topic in many branches of modern science and engineering. Within the visible range of electromagnetic wave, it is quite difficult to directly obtain phase information by the existing photodetectors. Phase retrieval provides an effective method to “figure out” the phase information from the captured intensity information, and has achieved successful applications in several scientific fields including astronomical observation, biomedical imaging and digital signal restoration. Algorithm is not only the core of phase retrieval, but is also the key to its development and applications. This paper demonstrates the basic principles of phase retrieval algorithms in combination with physical principles and signal processing methods, summarizes the development of various kinds of algorithms as well as their advantages and disadvantages, and briefly lists some typical applications in the field of optics. Finally, the challenges are pointed out, and the future development directions are described as: better convergence performance and noise robustness, phase-retrieval ability for more complex objects, compatibility for integration of multiple objectives and tasks.