The need for efficient normal integration methods is driven by several computer vision tasks such as shape-from-shading, photometric stereo, deflectometry. In the first part of this survey, we select the most important properties that one may expect from a normal integration method, based on a thorough study of two pioneering works by Horn and rooks (Comput Vis Graph Image Process 33(2): 174-208, 1986) and Frankot and Chellappa (IEEE Trans Pattern Anal Mach Intell 10(4): 439-451, 1988). Apart from accuracy, an integration method should at least be fast and robust to a noisy normal field. In addition, it should be able to handle several types of boundary condition, including the case of a free boundary and a reconstruction domain of any shape, i.e., which is not necessarily rectangular. It is also much appreciated that a minimum number of parameters have to be tuned, or even no parameter at all. Finally, it should preserve the depth discontinuities. In the second part of this survey, we review most of the existing methods in view of this analysis and conclude that none of them satisfies all of the required properties. This work is complemented by a companion paper entitled Variational Methods for Normal Integration, in which we focus on the problem of normal integration in the presence of depth discontinuities, a problem which occurs as soon as there are occlusions.