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In this expository paper, we consider the Hardy-Schr\"odinger operator$-\Delta -\gamma/|x|^2$ on a smooth domain \Omega of R^n with 0\in\bar{\Omega},and describe how the location of the singularity 0, be it in the interior of\Omega or on its boundary, affects its analytical properties. We compare thetwo settings by considering the optimal Hardy, Sobolev, and theCaffarelli-Kohn-Nirenberg inequalities. The latter rewrites:$C(\int_{\Omega}\frac{u^{p}}{|x|^s}dx)^{\frac{2}{p}}\leq \int_{\Omega} |\nablau|^2dx-\gamma \int_{\Omega}\frac{u^2}{|x|^2}dx$ for all $u\in H^1_0(\Omega)$,where \gamma <n^2/4, s\in [0,2) and p:=2(n-s)/(n-2). We address questionsregarding the explicit values of the optimal constant C, as well as theexistence of non-trivial extremals attached to these inequalities. Scaleinvariance properties lead to situations where the best constants do not dependon the domain and are not attainable. We consider two different approaches to"break the homogeneity" of the problem:  One approach was initiated by Brezis-Nirenberg and by Janelli. It is suitablefor the case where 0 is in the interior of \Omega, and consists of consideringlower order perturbations of the critical nonlinearity. The other approach wasinitiated by Ghoussoub-Kang , C.S. Lin et al. and Ghoussoub-Robert. It consistsof considering domains where the singularity is on the boundary.  Both of these approaches are rich in structure and in challenging problems.If 0\in \Omega, a negative linear perturbation suffices for higher dimensions,while a positive "Hardy-singular interior mass" is required in lowerdimensions. If the singularity is on the boundary, then the local geometryaround 0 plays a crucial role in high dimensions, while a positive"Hardy-singular boundary mass" is needed for the lower dimensions.

Sobolev inequalities for the Hardy-Schr\"odinger operator: Extremals and critical dimensions