Few questions arrive at the imam desk more often, and few are answered with more guilt than they need. Here is the fiqh, plainly.
Where the schools agree
The four schools agree that a prayer missed through sleep or forgetfulness is prayed when remembered, on the authority of the hadith: “Whoever sleeps through a prayer or forgets it, let them pray it when they remember it.”
Where they differ
They differ on prayers left deliberately: the majority hold that they are still made up alongside repentance, while a minority hold that the time cannot be recovered and repentance with abundant voluntary prayer is what remains.
A plan that survives contact with real life
Pair each missed prayer with a current one — one make-up prayer after each of the five — and keep a written count. A small, steady rate finishes; an ambitious one stops in a week.