Scholars distinguish between touching the physical mus-haf (the written Quran) and reciting its verses aloud or from memory. Regarding touching, the majority across the four schools hold that one must be in wudu to physically hold or touch the Arabic text of the mus-haf, based on Quran 56:79, 'None touch it except the purified.' A minority of scholars, including Ibn Hazm, read this verse as describing the angels who handle the preserved heavenly tablet rather than an operative legal ruling for ordinary believers, and permit touching without wudu, especially where unavoidable — for teaching young students, for instance, or with translations and Quran apps, which most contemporary scholars agree don't carry the same ruling as a physical mus-haf. Reciting Quran from memory without touching a physical copy does not require wudu at all — one may recite while walking, working, or otherwise without wudu. The point nearly all scholars agree on is that reciting is not permitted during major ritual impurity (janabah) until ghusl is performed; views differ on whether a menstruating woman may recite without touching the mus-haf itself.
Q&A · Purification
Do you need to be in a state of wudu to touch or recite the Quran?
Informational, not a personal fatwa. Consult a qualified scholar for rulings on your situation.