  We study the class of indecomposable two-dimensional Landau-Ginzburg theories
with (2,2) supersymmetry and central charge c < 6 with the aim of classifying
all such theories up to marginal deformations. Our results include cases
overlooked in previous classifications. The results are rigorous for three or
fewer fields and more generally are rigorous if we assume an extra bound.
Numerics suggest that we have the complete set of indecomposable
Landau-Ginzburg families with c<6. This set consists of 38 infinite families
and a finite list of 418 sporadic cases. The basic tools are classic results of
Kreuzer and Skarke on quasi-homogeneous isolated singularities and solutions to
certain feasibility integer programming problems.
