In the middle of 1980's, motivated by study of singularity theories, K. Saito introduced the notion of "elliptic root systems". Roughly speaking, they are root systems with two null directions. Furthermore, he had classified elliptic root systems $R$ with a one-dimensional subspace of $G$ of two dimensional null directions, under the assumption that the quotient affine root system $R/G$ is reduced. In our joint work with A. Fialowski and K. Iohara, we take off the assumption above, and give the classication of the pair $(R, G)$ with no assumption. In addition, we give an overview of the theory of elliptic root systems in this talk, and certain applications of this theory are also discussed.
