It is very important to update UK Visas and Immigration (UKVI) if there has been a change to certain changes of circumstance.
How you inform the UKVI that you have changed your address depends on whether your immigration permission is in the form of a biometric residence permit (BRP) or is in the form of a vignette (a sticker in your passport that you obtained before travelling to the UK).
Full information can be found on the UKVI website.
If there are any issues with your immigration permission, UKVI will normally write to you. If UKVI does not have your current address, you might not receive important correspondence from them that requires you to take action.
If you are a migrant living in the UK or your visa application was made in the UK, there is a form to:
UKCISA (The UK Council for International Student Affairs) has produced a table listing the various changes of circumstances that you need to notify UKVI about, and which form you need to use to do this. Click on the title 'Reporting changes of circumstances' on the link.
You may also be required to notify the UKVI of relevant changes to your studies. We have a separate webpage that explains this.