  Given a positive integer $n$ and a compact connected Riemann surface $X$, we
prove that the symmetric product $S^n(X)$ admits a Kaehler form of nonnegative
holomorphic bisectional curvature if and only if $\text{genus}(X) \leq 1$. If
$n$ is greater than or equal to the gonality of $X$, we prove that $S^n(X)$
does not admit any Kaehler form of nonpositive holomorphic sectional curvature.
