From the excellent Century of Flight website, a treasure trove for us airplane-lovers:

In the history of civil aviation, Germany may hold a unique distinction—that of having an airship service as its first airline, considered also by many as the world's first passenger airline. On November 16, 1909, German entrepreneurs created a company named DELAG (Deutsche Luftschiffahrt Aktien Gesellschaft). The company used one of the large airships built by Ferdinand Graf von Zeppelin, a retired military officer in his sixties. The early DELAG flights, between 1910 and 1914, and then after World War I, between 1919 and 1921, cost passengers between 100 and 200 reichsmarks, much more than the income of an average German worker. For the most part, DELAG airships carried wealthy foreigners across Germany to cities such as Berlin, Potsdam, and Dusseldorf.

Even though airships offered the first air transport services in Germany, their importance was slowly eclipsed by airplane service. In the late 1930s, after a series of spectacular accidents involving a number of airships—including the catastrophic explosion of the Hindenburg airship in May 1937—the Germans permanently discontinued airship transport in favour of airplanes.