  The paper aims to provide a full characterization of all operators $T\colon
\mathscr{P}(\mathbb{C}) \to \mathscr{P}(\mathbb{C})$ acting on the space of all
complex polynomials that satisfy the Leibniz rule \[ T(f\cdot g)= T(f)\cdot
g+f\cdot T(g) \] for all $f, g\in \mathscr{P}(\mathbb{C})$. We do not assume
the linearity of $T$. As we will see, contrary to the well-known theorems for
function spaces there are many other solutions here, not only differential
operators. From our main result, we also derive two corollaries, showing that
in some special cases operators that satisfy the Leibniz rule have some
particular form.
