You need to apply for a residence permit at your local Foreign Citizens' Office before your entry visa runs out. Citizens of Australia, Israel, Canada, New Zealand, the Republic of Korea, and the USA, who arrive without a visa and are planning to stay in Germany for a longer period of time, also need to apply for a residence permit within three months of entering the country.
We advise you to submit your application as soon as possible (approx. 8-10 weeks before your visa is due to expire). Please note that there are different types of visa and residence permits, that may require specific documents. You usually have to present the following documents at the Foreigners' Office (Ausländerbehörde):
Source: passages taken from EURAXESS
Goethe Welcome Centre
Biozentrum, Gebäude N100, Magistrale, EG, Raum 010
60438 Frankfurt am Main
For doctoral students: