  The question of determining the spatial geometry of the Universe is of
greater relevance than ever, as precision cosmology promises to verify
inflationary predictions about the curvature of the Universe. We revisit the
question of what can be learnt about the spatial geometry of the Universe from
the perspective of a three-way Bayesian model comparison. We show that, given
current data, the probability that the Universe is spatially infinite lies
between 67% and 98%, depending on the choice of priors. For the strongest prior
choice, we find odds of order 50:1 (200:1) in favour of a flat Universe when
compared with a closed (open) model. We also report a robust, prior-independent
lower limit to the number of Hubble spheres in the Universe, N_U > 5 (at 99%
confidence). We forecast the accuracy with which future CMB and BAO
observations will be able to constrain curvature, finding that a cosmic
variance limited CMB experiment together with an SKA-like BAO observation will
constrain curvature with a precision of about sigma ~ 4.5x10^{-4}. We
demonstrate that the risk of 'model confusion' (i.e., wrongly favouring a flat
Universe in the presence of curvature) is much larger than might be assumed
from parameter errors forecasts for future probes. We argue that a 5-sigma
detection threshold guarantees a confusion- and ambiguity-free model selection.
Together with inflationary arguments, this implies that the geometry of the
Universe is not knowable if the value of the curvature parameter is below
|Omega_curvature| ~ 10^{-4}, a bound one order of magnitude larger than the
size of curvature perturbations, ~ 10^{-5}. [abridged]
