When a two-dimensional area-preserving map f satisfies a property called the twist condition, there exists a function H for f, and the critical point of H can be used to determine the trajectory of the original map. In other words, we can formulate a "discrete variational problem" for f. In this talk, we discuss a method to show the existence of periodic orbits as well as heteroclinic and homoclinic orbits, which are characterized by asymptotic behavior, through discrete variational problems.