The surface-assisted polymerization and cyclodehydrogenation of specifically designed organic precursors provides a route toward atomically precise graphene nanoribbons, which promises to combine the outstanding electronic properties of graphene with a bandgap that is sufficiently large for room-temperature digital-logic applications. Starting from the basic concepts behind the on-surface synthesis approach, this report covers the progress made in understanding the different reaction steps, in synthesizing atomically precise graphene nanoribbons of various widths and edge structures, and in characterizing their properties, ending with an outlook on the challenges that still lie ahead.