This paper is dedicated to the study of the problem of existence of Poisson stable (Bohr/Levitan almost periodic, almost automorphic, almost recurrent, recurrent, pseudo periodic, pseudo recurrent and Poisson stable) motions of monotone sub-linear non-autonomous dynamical systems. The main results we establish in the framework of general non-autonomous (cocycle) dynamical systems. We apply our general results to the study of the problem of existence of different classes Poisson stable solutions of some types of non-autonomous evolutionary equations (Ordinary Differential Equations, Functional-Differential Equations with finite delay and Difference Equations).