  Is the Universe (a spatial section thereof) finite or infinite? Knowing the
global geometry of a Friedmann-Lema\^{\i}tre (FL) universe requires knowing
both its curvature and its topology. A flat or hyperbolic (``open'') FL
universe is {\em not} necessarily infinite in volume.
  Multiply connected flat and hyperbolic models are, in general, as consistent
with present observations on scales of 1-20{\hGpc} as are the corresponding
simply connected flat and hyperbolic models. The methods of detecting multiply
connected models (MCM's) are presently in their pioneering phase of development
and the optimal observationally realistic strategy is probably yet to be
calculated. Constraints against MCM's on ~1-4 h^{-1} Gpc scales have been
claimed, but relate more to inconsistent assumptions on perturbation statistics
rather than just to topology. Candidate 3-manifolds based on hypothesised
multiply imaged objects are being offered for observational refutation.
  The theoretical and observational sides of this rapidly developing subject
have yet to make any serious contact, but the prospects of a significant
detection in the coming decade may well propel the two together.
