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In this paper, we first establish the local well-posedness (existence,uniqueness and continuous dependence) for the Fornberg-Whitham equation in bothsupercritical Besov spaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p,r\leq+\infty$and critical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+\infty$, whichimproves the previous work \cite{y2,ho,ht}. Then, we prove the solution is notuniformly continuous dependence on the initial data in supercritical Besovspaces $B^s_{p,r},\ s>1+\frac{1}{p},\ 1\leq p\leq+\infty,\ 1\leq r<+\infty$ andcritical Besov spaces $B^{1+\frac{1}{p}}_{p,1},\ 1\leq p<+\infty$. At last, weshow that the solution is ill-posed in $B^{\sigma}_{p,\infty}$ with$\sigma>3+\frac{1}{p},\ 1\leq p\leq+\infty$.

The well-posedness, ill-posedness and non-uniform dependence on initial data for the Fornberg-Whitham equation in Besov spaces