  We presents the results of an extensive numerical study of the phase diagram
of the spin-1/2, \protect{$J_1$-$J_2$-$J_3$} Heisenberg model on a square
lattice, for both ferromagnetic and antiferromagnetic nearest-neighbor
interactions $J_1$, using exact diagonalization with periodic and twisted
boundary conditions. Comparison is made with published spin wave calculations.
We show that quantum fluctuations play a very important role, changing both the
extent and the wave vector of classical spiral phases, and leading to new
quantum phases where the classical spiral states have a high degeneracy. These
include a new phase with small or vanishing spin-stiffness, in addition to
known valence-bond-solid and bond-nematic phases.
