  The initial value problem $u(x,y,0)=u_0(x,y)$ for the Novikov-Veselov
equation $$\partial_tu+(\partial ^3 + \overline{\partial}^3)u +3(\partial
(u\overline{\partial}^{-1}\partial
u)+\overline{\partial}(u\partial^{-1}\overline{\partial}u))=0$$ is investigated
by the Fourier restriction norm method. Local well-posedness is shown in the
nonperiodic case for $u_0 \in H^s(\mathbb{R}^2)$ with $s > - \frac{3}{4}$ and
in the periodic case for data $u_0 \in H^s_0(\mathbb{T}^2)$ with mean zero,
where $s > - \frac{1}{5}$. Both results rely on the structure of the
nonlinearity, which becomes visible with a symmetrization argument.
Additionally, for the periodic problem a bilinear Strichartz-type estimate is
derived.
