  The paper describes a precise measurement of electron scattering off the
proton at momentum transfers of $0.003 \lesssim Q^2 \lesssim 1$\ GeV$^2$. The
average point-to-point error of the cross sections in this experiment is $\sim$
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 $Q^2$ values are used for a new determination of the
electric and magnetic radii. An empirical determination of the
Two-Photon-Exchange (TPE) 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.
