  We consider the extremal shot noise defined by
$$M(y)=\sup\{mh(y-x);(x,m)\in\Phi\},$$ where $\Phi$ is a Poisson point process
on $\bbR^d\times (0,+\infty)$ with intensity $\lambda dxG(dm)$ and $h:\bbR^d\to
[0,+\infty]$ is a measurable function. Extremal shot noises naturally appear in
extreme value theory as a model for spatial extremes and serve as basic models
for annual maxima of rainfall or for coverage field in telecommunications. In
this work, we examine their properties such as boundedness, regularity and
ergodicity. Connections with max-stable random fields are established: we prove
a limit theorem when the distribution $G$ is heavy-tailed and the intensity of
points $\lambda$ goes to infinity. We use a point process approach strongly
connected to the Peak Over Threshold method used in extreme value theory.
Properties of the limit max-stable random fields are also investigated.
