Are rules prescribing what sorts of clothes men and women can and cannot wear?
There is an explicit Torah commandment forbidding men to wear women's clothing and visa versa. I quote Deuteronomy 22:5:
לֹא יִהְיֶה כְלִי גֶבֶר עַל אִשָּׁה וְלֹא יִלְבַּשׁ גֶּבֶר שִׂמְלַת אִשָּׁה כִּי תוֹעֲבַת יְהֹוָה אֱלֹהֶיךָ כָּל עֹשֵׂה אֵלֶּה: A man's attire shall not be on a woman, nor may a man wear a woman's garment because whoever does these [things] is an abomination to the Lord, your God.
It is debated if this applies to wearing a full outfit of the opposite gender, or if even one article of clothing clearly belonging to the opposite gender is prohibited.
There are some exceptions, such as to protect from the rain, possibly in jest on Purim, and others.