It is well known that for a Tychonoff space X, the Stone-Cech compactification is the largest Hausdorff compactification of X. Clearly if a topological property P is hereditary and if X is not of class P, then X does not have a compactification with property P. But we may always ask when X has a Hausdorff compactification bX with the remainder bX\X belonging to a given class of spaces. In this talk we give a brief summary of such results as well as recent results obtained by Arhangel’skii.