  Let $X=\{X(t),t\in\mathbb{R}^N\}$ be a random field with values in
$\mathbb{R}^d$. For any finite Borel measure $\mu$ and analytic set
$E\subset\mathbb{R}^N$, the Hausdorff and packing dimensions of the image
measure $\mu_X$ and image set $X(E)$ are determined under certain mild
conditions. These results are applicable to Gaussian random fields,
self-similar stable random fields with stationary increments, real harmonizable
fractional L\'{e}vy fields and the Rosenblatt process.
