The need for efficient normal integration methods is driven by several computer vision tasks such as shape-from-shading, photometric stereo, deflectom-etry, etc. Our work is divided into two papers. In the present paper entitled Part I: A 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 Brooks [27] and by Frankot and Chellappa [19]. 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 both several types of boundary condition , including the case of a free boundary, and a domain of reconstruction 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 view of this analysis, we review most of the existing methods, and conclude that none of them satisfies all of the required properties. In the second paper entitled Part II: New Insights, we focus on the problem of normal integration in the presence of depth discontinuities, a problem which occurs as soon as there are occlusions.