The understanding of body measurements and body shapes in and between populations is important and has many applications in medicine, surveying, the fashion industry, fitness, and entertainment. Body measurement using 3D surface scanning technologies is faster and more convenient than measurement with more traditional methods and at the same time provides much more data, which requires automatic processing. A multitude of 3D scanning methods and processing pipelines have been described in the literature, and the advent of deep learning-based processing methods has generated an increased interest in the topic. Also, over the last decade, larger public 3D human scanning datasets have been released. This paper gives a comprehensive survey of body measurement techniques, with an emphasis on 3D scanning technologies and automatic data processing pipelines. An introduction to the three most common 3D scanning technologies for body measurement, passive stereo, structured light, and time-of-flight, is provided, and their merits w.r.t. body measurement are discussed. Methods described in the literature are discussed within the newly proposed framework of five common processing stages: preparation, scanning, feature extraction, model fitting, and measurement extraction. Synthesizing the analyzed prior works, recommendations on specific 3D body scanning technologies and the accompanying processing pipelines for the most common applications are given. Finally, an overview of about 80 currently available 3D scanners manufactured by about 50 companies, as well as their taxonomy regarding several key characteristics, is provided in the Appendix.