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How much notice do I have to give?

Assured Shorthold Tenancy

If you have an assured shorthold tenancy then you have no legal need to give notice at the end of a fixed term tenancy - although it would obviously be polite to do so.  Many landlords dispute this, but this document by Shelter on the Housing Law Practitioners Association website outlines the law involved (MS Word Reqd). 
Once the fixed term has passed - even by just one day - you have a 'Statutory Periodic Tenancy' under section 5 of the 1988 Housing Act.  Written notice required from tenants is a minimum of one month - with the leaving date being the last day of a tenancy period. 
With a monthly tenancy, this is easy, it is the same day of the month as your fixed term ended, for ASTs that ended on the 29th - 31st of the month, then the last day of the month applies if your final month doesn't have the right day in it.
With a weekly tenancy, it is also easy - the same day of the week as your tenancy ended.
Fortnightly, 4 weekly, 3 monthly tenancies will require a calender to work out the end of the tenancy period.
It is important to realise that tenancy periods may NOT coincide with rent days.  They often do, but do not have to.  For notice periods, it is the tenancy period that is relevant.  Although this document was written for a different purpose, it explains why tenancy periods are not dependant on rent days.  http://www.hlpa.org.uk/cms/wp-content/uploads/2011/09/FAQ-7-section-21-notices-last-day.docx
If you leave before the notice date shown above, you will be liable for rent until that date is reached - unless a new tenant moves in, in which case your liability ends the day before they move in.  In this case, any rent already paid may not be refundable.

Non-AST tenancies

Usually tenancies with an annual rent in excess of £100,000 per year.  These are simply governed by whatever your tenancy agreement states. 

Lodgers / Resident Landlords

If your landlord lives in the same property as you, then you are an 'excluded occupier'.  The legal requirement for notice is simply "reasonable notice", Usually, this would be the equivalent of one rent period.  However if you have a contract stating specific notice then this will be taken as 'reasonable notice' and you can not leave before the end of any period specified in your contract.