There is a myriad of external funding available for research and teaching purposes. Here are a few examples.
The German Research Foundation (DFG) provides funding through its many members from German universities, research institutions, scientific associations, and the Academies of Science and the Humanities.
The DAAD's extensive scholarship database provides funding opportunities for students, graduates, and academics.
The Alexander von Humboldt Foundation provides research fellowships and research awards for qualified scientists at all stages of their careers.
The German-American Fulbright Commission offers exchange programs for U.S. scholars who want to research and/or lecture at a German university or research institution.