Unipotent quantum groups are non-commutative versions of unipotent group schemes in algebraic geometry. Finite unipotent quantum groups only appear in positive characteristic. In this talk, we will provide a complete classification of such objects up to prime-cube dimension, which can be thought as a generalization of the well-known fact about the structure of p-groups of small orders. In contrast to the finiteness of isomorphism classes for each fixed order in group theory, we obtain nine infinite families among these prime-cube ones, which are all naturally parameterized by finite group quotients of the affine line. Further topics regarding representations, cohomology and invariant theory of small unipotent quantum groups will also be discussed during the talk.