Fixed-points theorems ensure existence of fixed-points of some functions, given that some conditions are satisfied. They have been widely used in approximation theory, dynamical systems, equilibrium theory, and so on. In this talk I will introduce some fixed-point models, and PPAD-completeness of the search version of discrete fixed-points, as well as their applications to fair division
problems, such as envy-free cake-cutting and consensus-halving problem.