Consider the problem of estimating the mean function underlying a set of noisy data. Least squares is appropriate if the error distribution of the noise is Gaussian, and if there is good reason to believe that the underlying function has some particular form. But what if the previous two assumptions fail to hold? In this regression setting, a robust method is one that is resistant against outliers, while a nonparametric method is one that allows the data to dictate the shape of the curve (rather than choosing the best parameters for a fit from a particular family). Although it is easy to find estimators that are either robust or nonparametric, the literature reveals very few that are both. In this thesis, a new method is proposed that uses the fact that the L\sb1 norm naturally leads to a robust estimator. In spite of the L\sb1 norm's reputation for being computationally intractable, it turns out that solving the least absolute deviations problem leads to a linear program with special structure. By utilizing this property, but over local neighborhoods, a method that is also nonparametric is obtained. Additionally, the new method generalizes naturally to higher dimensions; to date, the problem of smoothing in higher dimensions has met with little success. A proof of consistency is presented, and the results from simulated data are shown.