  A meromorphic connection on the complex projective line induces formal
connections at each singular point, and these formal connections constitute the
local behavior at the singularities. In this primarily expository paper, we
discuss the extent to which specified local behavior at singular points
determines the global connection. In particular, given a finite set of points
and a collection of ``formal types'' at these points, does there exist a moduli
space of meromorphic connections with this local behavior, and if so, when is
this moduli space nonempty or a singleton? In this paper, we discuss variants
of these problems (for example, the Deligne--Simpson and rigidity problems) as
the allowed singularities get progressively more complicated: first connections
with only regular singularities, next connections with additional unramified
irregular singularities allowed, and finally the general case.
