In response to complexity and diversity of a social phenomenon, the dispute also is various, therefore can not be settled efficiently by means of court adjudication to which applies a law strictly. To overcome such problems we are going to seek to make use of arbitration. According to Korean Arbitration Act Art. 3 (1), any dispute in private laws would be the object of arbitral proceedings. It could be the object of arbitral proceedings that disputes which are capable of a settlement by arbitration. It is a matter for debate that disputes containing punitive damages may be resolved by arbitration. This problem is concerning the arbitrability of the subject-matter of a dispute. To offer some solution to these issues, it is necessary to inquire into the nature of punitive damages. the policy and function of alimony, the fair apportionment of a loss. Moreover, international relations formed with international transactions should be considered. Punitive damages would be the object of arbitral proceedings as the dipute in private laws. When punitive damages pursue only punishment in the domestic arbitration that there is not foreign factors, arbitral tribunal could not make arbitral award containing punitive damages. However, if punitive damages are admitted under the rules applicable to substance of dispute, and there is the arbitration agreement in which is implied that the parties agree to submit to an arbitral award, arbitral tribunal could make arbitral award containing punitive damages in international arbitration. When it is questionable whether it is offend against our public policy or not, that we accept the effect of arbitral award containing punitive damages, and we admit the enforcement of it, we have to take the nature of punitive damages, the policy and function of alimony, the fair apportionment of a loss and the stability of international transactions into consideration.