Random walks on groups and harmonic functions on groups are intimately related to stationary group actions, which are a generalization of measure preserving group actions. An important invariant of stationary group actions is their Furstenberg Entropy. The Furstenberg Entropy realization problem is the question of determining the range of possible entropy values realizable for a given random walk. The talk will include an introduction to this field, an overview of what (little) is known, and some new results.