3-D Shapes

Aeneas

Please can someone tell me whether has managed to classify all possible 3-D shapes into a finite and usefully small number of categories? At school level, most solid shapes seem to be some part, or combination, of 3 categories:
1. irregular shapes.
2. prisms.
3. pyramids.
Even the sphere and all ployhedra can be taken care of in this way. However, there seem to be solid shapes that do not fit these categories. For example, a piece of wire whose cross-section chages uniformly along its length from circular to an ellipse of a certain eccentricity. Is it possible, then, to classify all possible shapes? If it hasn't been done, is there any logical reason why it can't be done?

(p.s. Sorry about the incomplete version of this. I pressed the wrong button and it got posted and I couldn't see how to delete it.)