  We present the two- and three-pion exchange contributions to the
nucleon-nucleon interaction which occur at
next-to-next-to-next-to-next-to-leading order (N4LO, fifth order) of chiral
effective field theory, and calculate nucleon-nucleon scattering in peripheral
partial waves with L>=3 using low-energy constants that were extracted from
pi-N analysis at fourth order. While the net three-pion exchange contribution
is moderate, the two-pion exchanges turn out to be sizeable and prevailingly
repulsive, thus, compensating the excessive attraction characteristic for NNLO
and N3LO. As a result, the N4LO predictions for the phase shifts of peripheral
partial waves are in very good agreement with the data (with the only exception
of the 1F3 wave). We also discuss the issue of the order-by-order convergence
of the chiral expansion for the NN interaction.
