  An \emph{additive coloring} of a graph $G$ is an assignment of positive
integers $\{1,2,...,k\}$ to the vertices of $G$ such that for every two
adjacent vertices the sums of numbers assigned to their neighbors are
different. The minimum number $k$ for which there exists an additive coloring
of $G$ is denoted by $\eta (G)$. We prove that $\eta (G)\leqslant 468$ for
every planar graph $G$. This improves a previous bound $\eta (G)\leqslant 5544$
due to Norin. The proof uses Combinatorial Nullstellensatz and coloring number
of planar hypergrahs. We also demonstrate that $\eta (G)\leqslant 36$ for
3-colorable planar graphs, and $\eta (G)\leqslant 4$ for every planar graph of
girth at least 13. In a group theoretic version of the problem we show that for
each $r\geqslant 2$ there is an $r$-chromatic graph $G_{r}$ with no additive
coloring by elements of any Abelian group of order $r$.
