The problem that I would like to talk about started about 20 years ago when Givental discovered that the Hirota bilinear equations of the KdV hierarchy can be described in terms of the periods of A_1 singularity. Shortly afterwards, myself and Givental were able to prove that the construction can be extended to all simple singularities which gave a description in terms of period integrals of the so-called principal Kac-Wakimoto hierarchies of ADE type. The problem is how far can we go beyond simple singularities or equivalently beyond simple Lie algebras? Can we apply the same ideas to the period integrals appearing in mirror symmetry and construct integrable hierarchies that have applications to enumerative geometry? This question is still not easy to answer but nevertheless there were several interesting developments during the years. I would like to talk about them.