  Let $\Omega$ be a bounded domain of $\mathbb{R}^{N}$, and $Q=\Omega
\times(0,T).$ We consider problems\textit{ }of the type % \[ \left\{
\begin{array} [c]{l}% {u_{t}}-{\Delta_{p}}u\pm\mathcal{G}(u)=\mu\qquad\text{in
}Q,\\ {u}=0\qquad\text{on }\partial\Omega\times(0,T),\\
u(0)=u_{0}\qquad\text{in }\Omega, \end{array} \right. \] where ${\Delta_{p}}$
is the $p$-Laplacian, $\mu$ is a bounded Radon measure, $u_{0}\in
L^{1}(\Omega),$ and $\pm\mathcal{G}(u)$ is an absorption or a source term$.$ In
the model case $\mathcal{G}(u)=\pm\left\vert u\right\vert ^{q-1}u$ $(q>p-1),$
or $\mathcal{G}$ has an exponential type. We prove the existence of
renormalized solutions for any measure $\mu$ in the subcritical case, and give
sufficient conditions for existence in the general case, when $\mu$ is good in
time and satisfies suitable capacitary conditions.
