  We study the Obata equation with Robin boundary condition $\frac{\partial
f}{\partial \nu}+af=0$ on manifolds with boundary, where $a \in
\mathbb{R}\setminus\{0\}$. Dirichlet and Neumann boundary conditions were
previously studied by Reilly \cite{R}, Escobar \cite{Es} and Xia \cite{X}.
Compared with their results, the sign of $a$ plays an important role here. The
new discovery shows besides spherical domains, there are other manifolds for
both $a>0$ and $a<0$. We also consider the Obata equation with non-vanishing
Neumann condition $\frac{\partial f}{\partial \nu}=1$.
