  Given a family of graphs $\mathcal{F}$ and an integer $r$, we say that a
graph is $r$-Ramsey for $\mathcal{F}$ if any $r$-colouring of its edges admits
a monochromatic copy of a graph from $\mathcal{F}$. The threshold for the
classic Ramsey property, where $\mathcal{F}$ consists of one graph, was located
in the celebrated work of R\"odl and Ruci\'nski.
  In this paper, we offer a twofold generalisation to the R\"odl--Ruci\'nski
theorem. First, we show that the list-colouring version of the property has the
same threshold. Second, we extend this result to finite families $\mathcal{F}$,
where the threshold statements might also diverge. This also confirms further
special cases of the Kohayakawa--Kreuter conjecture. Along the way, we supply a
short(-ish), self-contained proof of the $0$-statement of the
R\"odl--Ruci\'nski theorem.
