The edges of graphene nanoribbons (GNRs) have attracted much interest due to their potentially strong influence on GNR electronic and magnetic properties. Here we report the ability to engineer the microscopic edge termination of high-quality GNRs via hydrogen plasma etching. Using a combination of high-resolution scanning tunneling microscopy and first-principles calculations, we have determined the exact atomic structure of plasma-etched GNR edges and established the chemical nature of terminating functional groups for zigzag, armchair, and chiral edge orientations. We find that the edges of hydrogen-plasma-etched GNRs are generally flat, free of structural reconstructions, and terminated by hydrogen atoms with no rehybridlzation of the outermost carbon edge atoms. Both zigzag and chiral edges show the presence of edge states.