Further to the answers above, here are various quotes from the HMRC website:
Everyone who pays tax should, at some stage, receive a record of the tax they have paid. [Since] April 1996, the law requires you to keep those records and other records so that you can complete a tax return fully and accurately if you are asked to do so.
Everyone must by law keep records of their income, capital gains and expenditure for at least 22 months after the end of the tax year to which they relate, so that they can fill in a Tax Return fully and accurately if they get one.
People who have businesses, such as the self-employed, partners and those letting property, must keep their records for at least five years and ten months after the end of the tax year to which they relate.
It's helpful to keep information you get from
● your employer, if you have - or had - one, about your pay (including bonuses) and tax deducted, benefits in kind, expenses payments, and any share scheme arrangements ● the Benefits Agency about your state pension or other taxable Social Security benefits ● banks and building societies about the interest on your account(s).
Always keep you P60's and other tax documents. Wage slips for 3 years or so, but you can always get a duplicate copy of bank statements so you can get rid of those (I'd strongly advise a shreadder for that, I shread anything with my name and address on it.
Personally I keep everything in a filling draw. This is "archived" into folders periodically, but thats just me.