Josef Peter Fleischmann studied at the Technical University of Munich. He was active in WARR, the university's student rocketry group. WARR students had built their own engines and a suborbital rocket before Isar Aerospace existed.
In 2018 Fleischmann, Daniel Metzler and Markus Brandl founded Isar Aerospace near Munich. Fleischmann was the CTO, in charge of the technology of the Spectrum launch vehicle.
On 30 March 2025 the first Spectrum flight from Andoya in Norway ended after 30 seconds. On 5 September 2026 the second flight reached orbit.
In September 2026 Fleischmann stepped down as CTO to start Bavaria Instruments. He kept his stake in Isar Aerospace.