I wish to make a sweep of a profile (from a sketch) along a 3D curve (as opposed to a planar one). How do I perform this operation?

Create your profile as a sketch on an arbitrary plane. Then create your 3D curve, either by coordinate input or through points, and this should give you a Line Body. Ideally the 3D curve should start at the plane of the sketch, and start approximately normal to it, although I dont think it is essential. Then you have to make a Named Selection (from the Tools menu) and select the Line Body. Now when you set up a sweep operation, choose the sketch as the profile, and the Named Selection as the path, and it should work. You have to be a little careful about the alignment and curvature of the path as the profile cannot overlap back upon itself and create some sort of non-manifold surface. The operation will just fail in that case.

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