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There is a widespread custom to wear costumes on Purim. This practice is mentioned already by the Rama (Rabbi Moshe Isserles, Poland, 1525-1572), and thus has a clear, explicit basis in Halacha.
Numerous different explanations have been offered for the reason underlying this custom. One approach, perhaps, emerges from the Gemara’s comment in Masechet Megila in explaining why God subjected the Jews to the terror of Haman’s edict. The Gemara gives two reasons, including the fact that many years earlier, the Jews in Babylonia had bowed to the idol built at the command of the emperor Nebuchadnesar. The Jews did not intend to worship the statue, but were nevertheless guilty of giving an outward expression of idol worship. God therefore punished the Jewish people by making an outward appearance that they would be annihilated. He never intended for Haman to succeed in his plan to kill the Jews; it was rather arranged that it would appear as though they would be killed, to atone for the Jews’ giving the outward appearance of worshipping an idol. We therefore commemorate the Purim miracle by wearing masks, giving an outward appearance that is different from our true appearance.
Is it permissible for a person to dress up on Purim as a member of the opposite gender? May a man dress up as a woman, or a woman dress up as a man, on Purim?
This question was addressed by the Mahari Mintz, who wrote that he saw many people dress as members of the opposite gender on Purim in the presence of leading Hachamim, and the Hachamim did not object. He therefore concludes that although wearing the clothing of the opposite gender is explicitly forbidden by the Torah (Debarim 22:5), the Rabbis allowed this practice for the purpose of the Purim festivities. The Mahari Mintz compares this practice to the custom he observed of allowing children to grab candies from each other on Purim. Even though Halacha clearly forbids taking other people’s possessions even in jest, and considers this outright theft, in the context of the Purim celebration it is deemed permissible. Similarly, the Rama, in his Darcheh Moshe, records a custom to allow wearing on Purim clothes that contain Shaatnez on the level of Rabbinic enactment; these enactments were waived for the purpose of the special joy of Purim. Likewise, the Mahari Mintz writes, cross dressing was permitted on Purim as part of the holiday festivities.
However, there are a number of Rishonim (Medieval Halachic scholars), including the Rambam and Rabbi Eliezer of Metz, who write explicitly that dressing as a member of the opposite gender is forbidden under all circumstances, without exception. In their view, this is forbidden even if it is done only temporarily and as a joke. The Rambam, for example, decries the custom to cross dress at weddings as part of the festivities. It stands to reason that the Mahari Mintz would not have ruled as he did if he had seen the comments of these Rishonim. As for the custom to allow children to grab each other’s candies, this practice does not provide a basis to allow cross dressing on Purim. The principle of "Hefker Bet Din Hefker" grants the Rabbinical Court the authority, when it deems it appropriate, to designate someone’s property as ownerless. Therefore, for the purpose of the Purim festivities, the courts designated the children’s snacks as ownerless to allow the children to grab each other’s treats for fun. The Rabbis certainly have no power to suspend the explicit Torah prohibition against wearing the clothes of the opposite gender.
Therefore, Hacham Ovadia Yosef rules that one may not dress up as a member of the opposite gender on Purim, or allow his sons to dress as girls or his daughters to dress as boys.
Summary: It is customary to wear costumes on Purim, and this custom has strong basis in Halacha. However, it is forbidden for men to dress up as women, and vice versa. Children, too, should not be allowed to dress up as the opposite gender.