  We construct Lipschitz $Q$-valued functions which approximate carefully
integral currents when their cylindrical excess is small and they are almost
minimizing in a suitable sense. This result is used in two subsequent works to
prove the discreteness of the singular set for the following three classes of
$2$-dimensional integral currents: area minimizing in Riemannian manifolds,
semicalibrated and spherical cross sections of $3$-dimensional area minimizing
cones.
