I moved to London at the beginning of January to start working at my new job. At the moment I got payed but I'm not sure if I should do anything related to taxes.
Is my employer going to take care of everything or am I supposed to fill any form?
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If you are an employee, then you'll get your P60 form around the end of the tax year (5th April). Usually (if you are a basic employee, e.g. not self-employed or similar) you don't have to do anything with that. More information on the gov's site.
Also you should get a payslip every time you are paid (either on paper or electronically), where you can check the status of your current tax contributions. Please note that because you started to work in January, you are only working for 3 months in the country before the end of the tax year, meaning you might get a higher take-home-pay, as the income tax is calculated on your whole income in the actual tax year, and not monthly.