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). DESI BAO provide robust measurements of the transverse co-moving 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 \(0.1 < z < 4.2\) . DESI BAO data alone are consistent with the standard flat \(\varLambda \) CDM cosmological model with a matter density \(\varOmega _{m} = 0.295\pm 0.015\) . Paired with a baryon density prior from Big Bang Nucleosynthesis and the acoustic angular scale from the cosmic microwave background (CMB), DESI requires \(H_0 = (68.52\pm 0.62)\)  km/s/Mpc. Extending the baseline model with a constant dark energy equation of state parameter w, DESI BAO alone is consistent with \(w=-1\) . In models with a time-varying dark energy equation of state parametrized 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 \(\varLambda \) CDM model at the 2.5, 3.5 or \(3.9\sigma \) levels for the addition of the Pantheon+, Union3, or DES-SN5YR supernova datasets respectively. For the flat \(\varLambda \) CDM model with the sum of neutrino mass \(m_{\nu }\) free, combining the DESI and CMB data yields an upper limit \(m_{\nu } <0.072\) eV at 95% confidence level for a \(m_{\nu } > 0\) eV prior. This neutrino-mass constraint is substantially relaxed if the background dynamics are allowed to deviate from flat \(\varLambda \) CDM.