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Large-scale inference for random spatial surfaces over a region using spatial process models has been well studied. Under such models, local analysis of the surface (e.g., gradients at given points) has received recent attention. A more ambitious objective is to move from points to curves, to attempt to assign a meaningful gradient to a curve. For a point, if the gradient in a particular direction is large (positive or negative), then the surface is rapidly increasing or decreasing in that direction. For a curve, if the gradients in the direction orthogonal to the curve tend to be large, then the curve tracks a path through the region where the surface is rapidly changing. In the literature, learning about where the surface exhibits rapid change is called wombling, and a curve such as we have described is called a wombling boundary. Existing wombling methods have focused mostly on identifying points and then connecting these points using an ad hoc algorithm to create curvilinear wombling boundaries. Such methods are not easily incorporated into a statistical modeling setting. The contribution of this article is to formalize the notion of a curvilinear wombling boundary in a vector analytic framework using parametric curves and to develop a comprehensive statistical framework for curvilinear boundary analysis based on spatial process models for point-referenced data. For a given curve that may represent a natural feature (e.g., a mountain, a river, or a political boundary), we address the issue of testing or assessing whether it is a wombling boundary. Our approach is applicable to both spatial response surfaces and, often more appropriately, spatial residual surfaces. We illustrate our methodology with a simulation study, a weather dataset for the state of Colorado, and a species presence/absence dataset from Connecticut. [ABSTRACT FROM AUTHOR]