They are points where three or more edges meet.
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No, not all three-dimensional shapes have vertices. A vertex is an angle point of any shape. Spheres do not have vertices.
The special features of 3D shapes are they all are 3D which means they are fat but 2D shapes are flat they can not stand up.
It is impossible to name and draw ALL 3D shapes - there are infinitely many possibilities.
There are infinitely many possible shapes. One such is a cube with one vertex sliced off.
3d shapes with plane faces are called polyhedra (singular = polyhedron). The minimum number of vertices is 4, there is no maximum.