The KMOS Redshift One Spectroscopic Survey (KROSS) is an ESO guaranteed time survey of 795 typical star-forming galaxies in the redshift range z=0.8-1.0 with the KMOS instrument on the VLT. In this paper we present resolved kinematics and star formation rates for 584 z~1 galaxies. This constitutes the largest near-infrared Integral Field Unit survey of galaxies at z~1 to date. We demonstrate the success of our selection criteria with 90% of our targets found to be Halpha emitters, of which 81% are spatially resolved. The fraction of the resolved KROSS sample with dynamics dominated by ordered rotation is found to be 83$pm$5%. However, when compared with local samples these are turbulent discs with high gas to baryonic mass fractions, ~35%, and the majority are consistent with being marginally unstable (Toomre Q~1). There is no strong correlation between galaxy averaged velocity dispersion and the total star formation rate, suggesting that feedback from star formation is not the origin of the elevated turbulence. We postulate that it is the ubiquity of high (likely molecular) gas fractions and the associated gravitational instabilities that drive the elevated star-formation rates in these typical z~1 galaxies, leading to the ten-fold enhanced star-formation rate density. Finally, by comparing the gas masses obtained from inverting the star-formation law with the dynamical and stellar masses, we infer an average dark matter to total mass fraction within 2.2$r_e$ (9.5kpc) of 65$pm$12%, in agreement with the results from hydrodynamic simulations of galaxy formation.
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