It is permitted for a menstruating woman to do everything she normally does except for:
- Performing acts of worship that require ṭahāra (ritual purity): the formal prayer (ṣalāh), fasting, circumambulating the Kaʿba and performing iʿtikāf (staying overnight at the mosque for worship).
- Sexual intercourse
- Touching books of Quran (but she can read Quran as mentioned below).
It is permitted for menstruating women to enter mosques, for example in order to attend lectures. For more details see this article.
Permitted acts of worship include supplication (duʿāʾ), remembrance (dhikr) and reading or listening to Quran. A menstruating woman is permitted to read Quran according to most modern scholars and the Mālikī school, while in the past there was disagreement on whether she is allowed to do so or not. However there is an issue with whether touching a book of Quran is permitted for her or not, and the Mālikī opinion is that she can touch it provided that there is something between her skin and the book (for example if she wears gloves). She can also use an electronic device (like a smartphone) to read the Quran. The UAE-based mufti Ahmed al Haddad mentions the Mālikī opinions mentioned above and the permissibility of using electronic devices to read Quran while menstruating (archived link to his fatwa). Archived link to a fatwa on IslamOnline (a website by the respected Egyptian scholar Yusuf al-Qaradawi) that mentions the permissibility of reading the Quran while menstruating (provided the Quran is not touched directly). The Saudi mufti Khaled Al Mosleh expresses the same opinions (archived link to his fatwa).
Reader Questions
Where does the rule "women on their period aren't allowed to touch the Quran" come from? I always found it weird, because there was no physical Quran during the Prophet's (pbuh) lifetime
The Quran tells us:
77. It is a noble Quran.
78. In a well-protected Book.
79. None can grasp it except the purified.
80. A revelation from the Lord of the Worlds. (Verses 56:77-80)
Verse 79 above could be interpreted in many different ways, but one is that one should not touch a book of Quran except in a state of a ṭahāra (ritual purity), which would mean that a person who is in a state of janāba (ritual impurity, for example during the time after sexual intercourse and before showering) or menstruation should not touch it. We cannot be entirely certain that verse 79 is referring to touching a physical book of Quran, but to be on the safe side, it is best to take this potential interpretation seriously.
An additional piece of evidence is the Prophet’s letter of advice to the people of Yemen, known as Ṣaḥīfat ʿAmr bin Ḥazm and included in Imam Mālik’s hadith collection al-Muwaṭṭaʾ, in which he tells them not to touch a book of Quran except in a state of ṭahāra, which would mean that a menstruating women should not touch it.
As for the existence of books of Quran during the Prophet’s lifetime 
, we know that there were scribes who would write down the Quran as it was revealed, and that people did possess written versions of certain chapters and sections of the Quran, so the reference is to these partial written collections of Quranic verses.