Phase retrieval and quantitative phase imaging are central subjects in optical measuring and imaging technologies. The most well-established method for obtaining quantitative phase is through interferometry. However, this class of methods relies heavily on the superposition of two beams with a high degree of coherence, and complex interferometric device, stringent requirement on the environmental stability, and associated laser speckle noise greatly limit its applications in the field of microscopic imaging. On a different note, as one of the typical phase retrieval approaches, the transport of intensity equation (TIE) provides a new non-interferometric way to access the quantitative phase information. In recent years, it has been extensively studied and remarkable advancements have been made in the fields of adaptive optics, X-ray diffraction imaging, electron microscopy, and optical microscopy. In this work, we will review the basic principles and some recent advances in TIE phase retrieval, including its solutions, axial intensity derivative estimation, partially coherent imaging and light field imaging, with emphasis on its applications in the field of quantitative phase microscopy. The challenging problems as well as future research directions will also be discussed.