A well known result by Lagarias and Ziegler states that there are finitely many equivalence classes of d-dimensional lattice polytopes having volume at most K, for fixed constants d and K. We describe an algorithm for the complete enumeration of such equivalence classes for arbitrary constants d and K. The algorithm, which gives another proof of the finiteness result, is implemented for small values of K, up to dimension six. The resulting database contains and extends several existing ones, and has been used to correct mistakes in other classifications. When specialized to three-dimensional smooth polytopes, it extends previous classifications by Bogart et al., Lorenz and Lundman. Moreover, we give a structure theorem for smooth polytopes with few lattice points that proves that they have a quadratic triangulation and that we use, together with the classification, to describe smooth polytopes having small volume in arbitrary dimension. In dimension three we enumerate all the simplices having up to 11 interior lattice points and we use them to conjecture a set of sharp inequalities for the coefficients of the Ehrhart h*-polynomials, unifying several existing conjectures. Finally, we extract and discuss minimal interesting examples from the classification, and we study the frequency of properties such as being spanning, very ample, IDP, and having a unimodular cover or triangulation. In particular, we find the smallest polytopes which are very ample but not IDP, and with a unimodular cover but without a unimodular triangulation.