  We have investigated the quantum $J_1-J_2-J_3$ model on the honeycomb lattice
with exact diagonalizations and linear spin-wave calculations for selected
values of $J_{2}/J_{1}$, $J_{3}/J_{1}$ and antiferromagnetic ($J_{1}>0$) or
ferromagnetic ($J_{1}<0$) nearest neighbor interactions. We found a variety of
quantum effects: "order by disorder" selection of a N{\'e}el ordered
ground-state, good candidates for non-classical ground-states with dimer long
range order or spin-liquid like. The purely antiferromagnetic Heisenberg model
is confirmed to be N{\'e}el ordered. Comparing these results with those
observed on the square and triangular lattices, we enumerate some conjectures
on the nature of the quantum phases in the isotropic models.
