Merkel flew in especially to meet the new pontiff, an encounter that lasted 45 minutes -- unusually long for a private papal audience.
"These discussions covered the long relationship between the Vatican and Germany as well as common concerns including the social, political, economic and religious situation in Europe and the world," the Vatican said.
Merkel, the daughter of a Lutheran minister, attended the March 19 inauguration of Benedict XVI's successor but this was her first private meeting with Francis.
Merkel gave the pope -- who spent time in Frankfurt as a philosophy and theology student in the 1980s -- three volumes of poetry by the German poet Friedrich Hoelderlin and 107 CDs featuring the composer and conductor Wilhelm Furtwaengler -- both favourites of Francis.
The German leader joked she hoped the pontiff "would have the time to listen to them all".