BERLIN — What is the role of a Jewish museum? Is it for the Jewish community, or those non-Jews around it? And must its leader be Jewish?

These questions are swirling around the future of the Jewish Museum Berlin, one of the city’s most popular visitor attractions, after the abrupt departure last month of its director, Peter Schäfer. He left after a string of controversies in which critics — including the Israeli government and the main organization representing Jews in Germany — said the institution had gone beyond its mission and become overly political.

In particular, a recent exhibition about Jerusalem was seen as pro-Palestinian and anti-Israel. Other controversial exhibitions from Mr. Schäfer’s tenure dealt with attitudes toward circumcision and women’s head coverings, among other themes.

The museum, housed in a landmark zinc-clad zigzag building by the architect Daniel Libeskind, now finds itself caught between opposing sides as it seeks a new leader.