  We establish abstract limit theorems which provide sufficient conditions for
a sequence $(A_{l})$ of rare events in an ergodic probability preserving
dynamical system to exhibit Poisson asymptotics, and for the consecutive
positions inside the $A_{l}$ to be asymptotically iid (spatiotemporal Poisson
limits). The limit theorems only use information on what happens to $A_{l}$
before some time $\tau_{l}$ which is of order $o(1/\mu(A_{l}))$. In particular,
no assumptions on the asymptotic behavior of the system akin to classical
mixing conditions are used. We also discuss some general questions about the
asymptotic behaviour of spatial and spatiotemporal processes, and illustrate
our results in a setup of simple prototypical systems.
