Summary Functional data analysis involves the extension of familiar statistical procedures such as principal components analysis, linear modelling, and canonical correlation analysis to data where the raw observation xi is a function. An essential preliminary to a functional data analysis is often the registration or alignment of salient curve features by suitable monotone transformations hi of the argument t, so that the actual analyses are carried out on the values xi{hi(t)}. This is referred to as dynamic time warping in the engineering literature. In effect, this conceptualizes variation among functions as being composed of two aspects: horizontal and vertical, or domain and range. A nonparametric function estimation technique is described for identifying the smooth monotone transformations hi, and is illustrated by data analyses. A second-order linear stochastic differential equation is proposed to model these components of variation.