Wyo. Stat. § 5-9-142
In a civil matter no execution on a judgment may be stayed unless and until the party applying for the appeal, or someone for him, gives bond in an amount and with sufficient surety approved by the judge of the district court or the clerk of the district court, to abide by such judgment as may be rendered on appeal if such appeal is perfected, or if not so perfected, then to satisfy the judgment of the circuit court. If such bond is furnished by or on behalf of any party against whom the judgment has been rendered for money or property or both, the bond shall be conditioned for the performance and satisfaction of such judgment or order as may be entered against such party on appeal, and for the payment of all costs and damages which may be awarded against him in the district court. If the appeal is by a party against whom there is no recovery except for costs, the bond shall be conditioned for the payment of such costs and damages as may be awarded against him on the appeal.
W.S. 5-9-142