The discovery of solitons and completely integrable partial differential equations (PDEs) constitutes a paradigm in mathematical analysis and modern physics. Its impact on the development of our understanding of PDEs in physics (both classical and quantum), pure analysis, and differential geometry can hardly be overrated. In this proposal, we present a research agenda that aims at a new class of completely integrable PDEs, which in addition to having a Lax pair structure with an infinite hierarchy of conservation laws, also features a nonlinear part of critical type. In contrast to -- by now classical -- integrable equations such as the Kortweg-deVries, Benjamin-Ono and cubic NLS, the common presence of integrable structures and criticality lead to new phenomena such as wave turbulence (in the sense of growth of Sobolev norms) or even blowup of solutions. Therefore, our main points of interest for future research are directed towards a deeper insight into the mathematical interplay between features of complete integrability and critical behavior.