British and Irish citizens (but not British "nationals" from Hong Kong or other places overseas) can fly within the British Isles with a photocard driving licence. Other EU citizens can use their national ID card.
Everyone else still needs a passport. You will probably need to produce a passport to check in, though.
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It all depends on the airline. British Airways does not require any photo ID to board. And if you use online check-in or self-service check in, then no photo ID at all is needed.
It has been reported that the Strathclyde Police sometimes ask for ID from passengers as they arrive.
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Assuming that Glasgow is in Scotland, and that Birmingham is in England rather than Alabama, then in principle, no. However, airlines are justified at any time in asking for proof of identity, and visitors from outside the EU are required to carry passports and visas at all times. So if you're a Brit, take whatever form of identification you normally use - which might well be a passport - and if you're not, take your passport.
On some airlines a passport (or other photo id) is needed, e.g. easyJet and flybe. They do this to check that the person bearing the ticket is the same one named on it. Airlines like BA and bmi do not need to do this.
Scotland and England are both members of the United Kingdom and their citizens hold British passports. Equally no other nationalities in the UK (Northern Irish and Welsh) need passports to travel between countries. Scotland and England are both members of the United Kingdom and their citizens hold British passports. Equally no other nationalities in the UK (Northern Irish and Welsh) need passports to travel between countries.
If you're flying from Aberdeen in Scotland to Bermingham in England then you won't need a passport, but the airline will probably ask for photo ID.
Check the airline's website to see what ID will be acceptable.