This paper describes a precise measurement of electron scattering off the proton at momentum transfers of 0.003≲Q2≲1 GeV2. The average point-to-point error of the cross sections in this experiment is ∼0.37%. These data are used for a coherent new analysis together with all world data of unpolarized and polarized electron scattering from the very smallest to the highest momentum transfers so far measured. The extracted electric and magnetic form factors provide new insight into their exact shape, deviating from the classical dipole form, and of structure on top of this gross shape. The data reaching very low Q2 values are used for a new determination of the electric and magnetic radii. An empirical determination of the two-photon-exchange correction is presented. The implications of this correction on the radii and the question of a directly visible signal of the pion cloud are addressed.