As a non-EU Citizen you can apply for a temporary visa extension (up to 3 months) to remain in Germany whilst your residence visa application is being processed, or simply because you want to spend additional time in Germany after your Schengen visa expires (90 days in every 180 days).
Tip: If you are visiting Hamburg, Germany or other EU-member states on a Schengen visitor visa and wish to stay in Germany for an additional 90 days after the end of your Schengen visa consider applying for a temporary visa extension from Welcome Center Hamburg.
You can pick up this short-term visa extension paper without appointment at the Welcome Center Hamburg. Bring your passport and the filled application form Application for the issue / renewal of a residence permit (available from Hamburg.de > Application forms), there are also blank applications available from the information desk.
The general requirement for this temporary visa extension is that you have submitted your residence visa application before requesting a temporary visa, this makes sense except where you are yet to submit your residence visa application
and need to travel abroad; in this case an extension can be approved in exceptional cases.
Update: You cannot travel outside of Germany, in any circumstance if your temporary visa extension has ‘Gilt nicht für Auslandsreisen’ (Not valid for travel abroad) printed on it. Mine did and because of this applied for a Working-Holiday visa (available to ‘best-friend’ nations of Germany for under 30 year olds) which allows for immediate international travel.
This visa extension doesn’t allow international travel unless stated otherwise so there may be issues at immigration when leaving or returning to Germany when traveling with this visa extension paper, therefore you are discouraged to travel outside of Germany unless absolutely necessary.
Update: See notice above, you cannot re-enter Germany after your Schengen visa has expired with this paper. You could leave Germany a day before your last Shengen visa calendar day and may be able to re-enter on the last calendar day. I’m not putting this one to the test… 😉