The problem of estimating a smooth fault line in a bivariate regression, or density surface, is of considerable importance in various applications, such as computerized edge detection or the biological and geological sciences. In this talk, we study properties of a new estimator for a fault line in both settings. This estimator is constructed by maximizing a likelihood in a locally-linear model for the edge. The approach offers a unifying thread to the two problems, which are usually considered separately from each other. The convergence rate of the estimator comes within the known minimax-optimal convergence rate by an arbitrarily small power of the design intensity. Our main focus is on investigation of the local behavior of this estimator, through which we obtain asymptotic confidence bands for the fault line, both pointwise and simultaneous. The pointwise distance between the fault line and its bias-corrected estimator has a distribution which equals that of the location of the maximum of a Gaussian process with quadratic drift, and thus resembles a commonly encountered limit of M-estimators. Several examples, using artificial data, illustrate finite-sample performance. This is joint work with Peter Hall, CMA, ANU