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. A determination of the geometry of the Universe (which can be either flat, closed or open)
would also allow to measure its size, which is finite only in the case of a closed Universe.
In this talk I will 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. After reviewing the status of current cosmological data sets, I will discuss how we can compute the probability that the Universe is
spatially infinite, and present results from a compilation of observational data. I will also discuss the future prospects for improving on current
bounds, arguing 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}.