We present cosmological results from the measurement of baryon acoustic oscillations (BAO) in galaxy, quasar and Lyman-alpha forest tracers from the first year of observations from the Dark Energy Spectroscopic Instrument (DESI), to be released in the DESI Data Release 1. DESI BAO provide robust measurements of the transverse comoving distance and Hubble rate, or their combination, relative to the sound horizon, in seven redshift bins from over 6 million extragalactic objects in the redshift range 0.1 < z < 4.2. To mitigate confirmation bias, a blind analysis was implemented to measure the BAO scales. DESI BAO data alone are consistent with the standard flat Lambda CDM cosmological model with a matter density Omega(m) = 0.295 +/- 0.015. Paired with a baryon density prior from Big Bang Nucleosynthesis and the robustly measured acoustic angular scale from the cosmic microwave background (CMB), DESI requires H-0 = (68.52 +/- 0.62) km s(-1) Mpc(-1). In conjunction with CMB anisotropies from Planck and CMB lensing data from Planck and ACT, we find Omega(m) = 0.307 +/- 0.005 and H-0 = (67.97 +/- 0.38) km s(-1) Mpc(-1). Extending the baseline model with a constant dark energy equation of state parameter w, DESI BAO alone require w = -0.99(-0.13)(+0.15). In models with a time-varying dark energy equation of state parametrised by w(0) and w(a), combinations of DESI with CMB or with type Ia supernovae (SN Ia) individually prefer w(0) > -1 and w(a) < 0. This preference is 2.6 sigma for the DESI+CMB combination, and persists or grows when SN Ia are added in, giving results discrepant with the Lambda CDM model at the 2.5 sigma, 3.5 sigma or 3.9 sigma levels for the addition of the Pantheon+, Union3, or DES-SN5YR supernova datasets respectively. For the flat Lambda CDM model with the sum of neutrino mass Sigma m(nu) free, combining the DESI and CMB data yields an upper limit Sigma m(nu) < 0.072 (0.113) eV at 95% confidence for a Sigma m(nu) > 0 (Sigma m(nu) > 0.059) eV prior. These neutrino-mass constraints are substantially relaxed if the background dynamics are allowed to deviate from flat Lambda CDM.