This thesis is focused on a more general type of optimal stopping problems in discrete time. Varying approaches of viewing this problem are discussed and introduced, e.g.\ using a space of couplings under linear constraints or so-called adapted random probability measures. A connection between these views is made and existence of an optimal solution is shown. Further, a modified version of Monge-Kantorovich duality is established. The final sections show a monotonicity principle with examples. For a special class of cost functions, optimality (and uniqueness) of a "greedy strategy" is established. In particular, the proof resembles the main idea behind a monotonicity principle for discrete time, which in turn is based on a monotonicity principle for continuous time. Finally, optimality of the "greedy strategy" is shown using monotonicity.