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Generally keep yourself covered; no mini-skirts, open-backs, bare knees or shoulders.
Especially in the Vatican Museums but in many churches in Italy, you will be asked to cover up if it is deemed that you are showing too much skin. The simple answer, especially on hot days, is to bring a shawl to put on when entering sacred spaces. Churches are generally a bit chilly compared to outside, so a shawl will keep you warm as well.
For more information, read about the dress regulations for the Colosseum and Ancient Rome tour.