The Radiation Assessment Detector (RAD), onboard the Mars Science Laboratory (MSL) rover Curiosity, measures the energetic charged and neutral particles and the radiation dose rate on the surface of Mars. An important factor for determining the biological impact of the Martian surface radiation is the specific contribution of neutrons, with their deeper penetration depth and ensuing high biological effectiveness. This is very difficult to measure quantitatively, resulting in considerable uncertainties in the total radiation dose. In contrast to charged particles, neutral particles (neutrons and gamma rays) are generally only measured indirectly. Measured spectra are a complex convolution of the incident particle spectrum with the detector response function and must be unfolded. We apply an inversion method (based on a maximum likelihood estimation) to calculate the neutron and gamma spectra from the RAD neutral particle measurements. Here we show the first spectra on the surface of Mars and compare them to theoretical predictions. The measured neutron spectrum (ranging from 8 to 740 MeV) translates into a radiation dose rate of 14±4μGy/d and a dose equivalent rate of 61±15μSv/d. This corresponds to 7% of the measured total surface dose rate and 10% of the biologically relevant surface dose equivalent rate on Mars. Measuring the Martian neutron and gamma spectra is an essential step for determining the mutagenic influences to past or present life at or beneath the Martian surface as well as the radiation hazard for future human exploration, including the shielding design of a potential habitat. Key Points We calculated the Martian neutron and gamma spectra We compare the results to Planetocosmics simulations We calculate dose and dose equivalent rate for the neutron spectrum