Can a Syrian citizen, who holds a USA green card, travel through Germany without a visa?
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The short answer is it depends. To transit in Germany, you do not need a visa (even though most Syrians do). To pass through Schengen border controls, however (that is, to leave the international zone of a German airport), you do need a visa.
Entering the Schengen zone
To enter the Schengen zone, which includes Germany, you need a short-stay visa. The green card does not exempt you from this requirement. This is governed by Regulation 539/2001, Annex I. In this sense, "entering" the zone means passing through passport control. Being present in the "international" zone of an airport (or sea port, for that matter) does not require a short-stay visa.
Transiting through a German airport
Some countries' citizens, however, require an "airport transit visa" to fly to a Schengen airport if they plan to remain in the international zone. For some countries' citizens, this requirement exists for the entire Schengen zone, but Syria is not one of those countries. Instead, Syria is on the list of countries where this requirement exists only in certain Schengen countries, and Germany is among the countries that have imposed an ATV requirement on Syrian citizens.
However, your green card exempts you from any requirement to hold an airport transit visa (Schengen Visa Code Annex V).
This applies only to transits where you fly to a Schengen airport from outside the Schengen area, remain in the international zone of the airport (for example, without needing to retrieve your checked luggage), and then board a flight to a destination outside the Schengen zone. If you need to fly between two airports that are both in the Schengen zone, you will need a short-stay visa. If you need to retrieve your luggage, you will need a short-stay visa. If you have an overnight layover in an airport that closes at night, you will need a short-stay visa.
More information is available on this site at Do I need a visa to transit (or layover) in the Schengen area?
The EU's page on the topic is Who must apply for a Schengen visa.