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Recent developments in geometric measure theory and harmonic analysis have led to new and deep results concerning the regularity of the support of measures which behave 'asymptotically' (for balls of small radius) as the Euclidean volume. A striking feature of these results is that they actually characterize flatness of the support in terms of the asymptotic behavior of the measure. Such characterizations have led to important new progress in the study of harmonic measure for non-smooth domains.This volume provides an up-to-date overview and an introduction to the research literature in this area. The presentation follows a series of five lectures given by Carlos Kenig at the 2000 Arkansas Spring Lecture Series at the University of Arkansas. The original lectures have been expanded and updated to reflect the rapid progress in this field. A chapter on the planar case has been added to provide a historical perspective. Additional background has been included to make the material accessible to advanced graduate students and researchers in harmonic analysis and geometric measure theory.