  It was recently pointed out that linear quantum fields $\hat \phi(x)$ can be
meaningfully propagated across the big bang (and the big crunch) singularities
of spatially flat Friedmann, Lema\^itre, Robertson, Walker (FLRW) universes
\cite{ADLS2021}. Recall that $\hat \phi(x)$, as well as renormalized
observables $\langle\hat \phi(x)^2 \rangle_{ren}$ and $\langle \hat
T_{ab}(x)\rangle_{ren}$, are distribution-valued already in Minkowskian quantum
field theories. It was shown that they can be extended as well-defined
distributions even when these space-times are enlarged to include the big-bang
(or the big crunch). We generalize these results to spatially closed and open
FLRW models, showing that this `tameness' of cosmological singularities is not
an artifact of the technical simplifications due to spatial flatness. Our
analysis also provides explicit expressions of $\langle\hat \phi(x) \hat
\phi(x') \rangle_{ren}$, $\langle\hat \phi(x)^2 \rangle_{ren}$ and $\langle
\hat T_{ab}(x)\rangle_{ren}$ in closed and open universes for minimally coupled
massless scalar fields and discuss the ambiguities in the definition of
$\langle \hat T_{ab}(x)\rangle_{ren}$ at the big-bang. While the technical
expressions are more complicated than in the spatially flat case, there is also
an unexpected conceptual simplification: the infrared divergence \cite{fp} is
now absent because, in effect, the spatial curvature provides a natural cutoff.
Finally, we further clarify the sense in which quantum field theory can
continue to be well defined even though the extended space-time is not globally
hyperbolic because of the singularity, and suggest directions for further work.
