  We study a class of Dirichlet boundary value problems whose prototype is
\begin{equation}\label{1.2abs}
  \left\{\begin{array}{ll}
  -\Delta_p u =h(u)|\nabla u|^p+u^{q-1}+f(x)\, &\quad\hbox{in } \ \Omega\,,\\
u\ge 0\,,&{\quad\hbox{in } \ \Omega}\\ u = 0\,&\quad\hbox{on }\partial
\Omega\,,\end{array}\right. \end{equation} where $\Omega$ an open bounded
subset of $\mathbb R^N$, $0<q<1$, $1<p<N$, $h$ is a continuous function and $f$
belongs to a suitable Lebesgue space. The main features of this problem are the
presence of a singular term and a first order term with natural growth in the
gradient. A priori estimates and existence results are proved depending on the
summability of the datum $f$.
