We study several graphene devices able to generate non-linear effects in the current─voltage characteristics and in particular negative differential resistance (NDR) effects. This theoretical investigation is based on numerical charge transport simulation in the Green's function approach applied to a tight-binding Hamiltonian for particles in graphene. Depending on the device, the physical mechanism involved in the NDR effect may be different: (i) the mismatch of modes between left and right sides of a P+/P zigzag ribbon junction, (ii) the modulation of interband tunnelling in P/N junctions (tunnel diodes and tunnel field-effect transistors) or (iii) the modulation of chiral tunnelling in ‘conventional’ graphene transistors. We emphasize the advantages of exploiting different approaches of bandgap engineering in the form of graphene nanoribbons (GNRs) or nanomesh lattices (GNM), the latter resulting from a periodic array of nanoholes in graphene sheets. In particular, such nanostructuring allows us to design position-dependent bandgaps in devices, which is shown to make possible the optimization of device operation and, here, to get very high peak-to-valley ratio of the NDR. In the case of GNR nanostructuring, it is shown that appropriate bandgap engineering can even make the current─voltage characteristics of tunnel diodes weakly sensitive to the atomic edge disorder. Finally, GNM lattices are shown to be a very promising way to open large bandgaps in wide sheets of graphene and to introduce bandgaps locally with a view to optimizing the device operation and performance.