PhD scholarships (BMS Phase II, RTG's, IRTG's etc.):

Duration: 2+1 years. The extension is subject to the success of the first two years.

Professional obligations: Students have to attend the research seminars and summer schools of the corresponding institute, and to follow the PhD regulations of the corresponding university. Usually there are no teaching obligations for PhD students with scholarship. Students are expected to finish their PhD thesis in 3 years; it depends on the type of scholarship if extension is possible or not.

Monthly income: The stipend is 1468€ per month (regulated by the state of Germany).

Taxes, health insurance: The scholarships are tax-free, health insurance has to be paid separately. There is no official working time.

Holidays: Holidays can be taken according to agreement with the supervisor.

PhD positions (DFG's, SFB's etc.):

Duration: The positions are also typically given for 2+1 years.

Professional obligations: Usually similar to the case of the scholarships.

Monthly income: The salary that you actually obtain is similar to the amount of the PhD scholarship, 1400-1500€. However, usually you will have a slightly higher income because you don't have to subtract the health insurance. If your job is paid by the German state, you get some additional salary (tax reimbursement) in each November.

Taxes, health insurance: The main difference is that if you have a position, you automatically pay taxes, pension contribution and health insurance to the state of Germany. Paying taxes and pensions in Germany may be useful for later.

Holidays: The amount of holidays is regulated, it is usually up to 6 weeks per year.

Industry-based research positions (ZIB etc.) and teaching positions at the universities:

Duration: Depends on your obligations apart from your own research, typically up to 5 years. During this period, you can continue being a member of BMS.

Professional obligations: Such a job usually give you various working tasks outside doing your research, e.g. teaching or programming. This results to a higher salary, a lower amount of holidays (typically 4 weeks per year), and a slower progress of your PhD thesis.

Monthly income: At least 1400-1500€, sometimes even 2000€ if you have a full position (especially in computer science).

Taxes, health insurance: See the case of PhD positions.

Holidays: They are regulated. In the case of full positions, at least 4 weeks, in the case of 3/4 positions, typically up to 6 weeks.

BMS gap funding: applicable if your external funding starts later than the end of your Phase I scholarship, or if there is a lack of funding for several months in the middle of your Phase II studies. In principle, one should not expect more than the Phase I monthly income (1000€).

If you are eligible for BMS Phase II scholarship according to your Phase I results, you may get the whole BMS Phase II scholarship for a part of your PhD studies (e.g. if it is known that external funding will be available one year later).

DAAD gives scholarships to students according to their country. Their website is available here: https://www.daad.de/en/. A useful application on the website shows you if you can apply for a DAAD scholarship in Mathematics in Germany: https://www.daad.de/deutschland/studienangebote/international-programs/en/.