The field of frustrated magnetism has been enriched significantly by the
discovery of various kagome lattice compounds. These materials exhibit a great
variety of macroscopic behaviours ranging from magnetic orders to quantum spin
liquids. Using large-scale exact diagonalization, we construct the phase
diagram of the $S=1/2$ $J_1$-$J_2$ kagome Heisenberg model with $z$-axis
Dzyaloshinskii-Moriya interaction $D_z$. We show that this model can
systematically account for many of the experimentally observed phases. Small
$J_2$ and $D_z$ can stabilize respectively a gapped and a gapless spin liquid.
When $J_2$ or $D_z$ is substantial, the ground state develops a $\mathbf{Q}=0$,
$120^\textrm{o}$ antiferromagnetic order. The critical strengths for inducing
magnetic transition are $D^c_z \sim 0.1\, J_1$ at $J_2=0$, and $J^c_2\sim 0.4\,
J_1$ at $D_z=0$. The previously reported values of $D_z$ and $J_2$ for
herbertsmithite [ZnCu$_3$(OH)$_6$Cl$_2$] place the compound in close proximity
to a quantum critical point.
