Principal curves are smooth one-dimensional curves that pass through the middle of a p-dimensional data set, providing a nonlinear summary of the data. They are nonparametric, and their shape is suggested by the data. The algorithm for constructing principal curves starts with some prior summary, such as the usual principal-component line. The curve in each successive iteration is a smooth or local average of the p-dimensional points, where the definition of local is based on the distance in arc length of the projections of the points onto the curve found in the previous iteration. In this article principal curves are defined, an algorithm for their construction is given, some theoretical results are presented, and the procedure is compared to other generalizations of principal components. Two applications illustrate the use of principal curves. The first describes how the principal-curve procedure was used to align the magnets of the Stanford linear collider. The collider uses about 950 magnets ...