  We give nontrivial bounds for the bilinear sums $$ \sum_{u = 1}^{U}
\sum_{v=1}^V \alpha_u \beta_v \mathbf{\,e}_p(u/f(v)) $$ where
$\mathbf{\,e}_p(z)$ is a nontrivial additive character of the prime finite
field ${\mathbb F}_p$ of $p$ elements, with integers $U$, $V$, a polynomial
$f\in {\mathbb F}_p[X] $ and some complex weights $\{\alpha_u\}$,
$\{\beta_v\}$. In particular, for $f(X)=aX+b$ we obtain new bounds of bilinear
sums with Kloosterman fractions. We also obtain new bounds for similar sums
with multiplicative characters of ${\mathbb F}_p$.
