half a sphere
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Technically, there's no shape with no surface and no vertices, but a sphere is close. It has no vertices but has just one surface.
A cone.
If you are a cube, one of the sides is called a face. Same with a cone, the flat part is a face.
A cone would fit the given description
An ovoid (egg-shape) is one possible answer. A smooth blob (to use a very technical term!) is another.