Phase is an important component of an optical wavefield bearing the information of the refractive index, optical thickness, or the topology of the specimen. Phase retrieval is a central problem in many areas of physics and optics since the phase of a wavefield is not accessible directly. The most well-established method for obtaining quantitative phase is through interferometry, such as digital holography. However, this class of methods relies on coherent illumination, therefore, plagued with problems of speckle that prevent the formation of high quality images. On a different note, quantitative phase can be retrieved by transport-of-intensity equation (TIE) using only object field intensities at multiple axially displaced planes. TIE has been increasingly investigated during recent years due to its unique advantages over interferometric techniques: it is non-interferometric, works with partially coherent illumination, computationally simple, no need to phase unwrapping, and does not require a complicated optical system. In this paper, we will review some recent new developments in TIE phase retrieval: including its numerical solution, treatment of boundary problem and the low-frequency artifacts, and configurations for dynamic phase imaging. We also reexamine TIE in terms of phase-space optics, demonstrating the effect of partially coherent illumination on phase reconstruction, and connecting it to light field imaging at the geometry optics limit.