BAföG, actually the abbreviation for "Bundesausbildungsförderungsgesetz" (Federal Training Assistance Act), is the state financing of studies in Germany.
Half of it consists of an interest-free loan and the other half of a grant - in other words, money is gifted to you. You have to pay back a maximum of 10,010 euros of your loan at the end of your training, or even less: depending on the amount, there are additional discounts for early repayment.
Whether and how much BAföG you can receive depends on numerous factors. Decisive factors include your assets and income, the income of your parents and the number and level of education of your siblings. A BAföG calculator (e.g. here) can give you a rough idea. If in doubt, it is always worthwhile to submit an application!
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