In this talk I give an overview on the existing results for two questions associated with a Hamiltonian function that is selected randomly according to a probability measure that is translation invariant (this includes Hamiltonian functions that are almost periodic in position or in both position and momentum) 1. Consider a Hamiltonian ODE with a Hamiltonian function that is periodic in time. As in Arnold's conjecture we may wonder whether or not such an ODE has periodic orbits. 2. Analogously, we may study Hamilton-Jacobi PDEs and examine the question of homogenization that is closely related to the long time behavior of the solutions to the corresponding Hamiltonian ODE.