Jazzfest Bonn has been running since 2010, with every single style represented. The annual festival is usually held in May. More information can be found at:
There are plenty of other opportunities to hear music, such as on various Sundays during the summer on the roof of the Art and Exhibition Hall of the Federal Republic of Germany (Kunst- und Ausstellungshalle der Bundesrepublik Deutschland) and on Friday evenings in the Rheinaue Park beer garden. There is also often jazz being played in the park as part of the "Musik im Park" series.