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I am aware of circles of curvature and I am simply wondering to what extent does this generalize to $n$-dimensions. Specifically, if some surface in $n$-dimensional space is represented parametricaly, how does one determine the $n$-sphere of curvature at any given point?

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The 3-sphere of curvature is also called an osculating sphere. –  Mike Spivey Sep 13 '11 at 4:00
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But note that this is a sphere in contact with a curve, not with a two-dimensional surface. In general, a (hyper)surface doesn't have a sphere of curvature but a (hyper)ellipsoid of curvature, since it can have different curvature in different directions. By the way, if a (hyper)surface in $n$-dimensional space had a sphere of curvature, it would be an $(n-1)$-sphere, not an $n$-sphere. –  joriki Sep 13 '11 at 8:07
    
Space curves do have osculating spheres and osculating circles (the circle formed by the intersection of the osculating sphere and osculating plane). –  J. M. Sep 13 '11 at 8:57

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