  We construct the Kuranishi spaces, or in other words, the versal
deformations, for the following classes of connections with fixed divisor of
poles $D$: all such connections, as well as for its subclasses of integrable,
integrable logarithmic and integrable logarithmic connections with a parabolic
structure over $D$ . The tangent and obstruction spaces of deformation theory
are defined as the hypercohomology of an appropriate complex of sheaves, and
the Kuranishi space is a fiber of the formal obstruction map.
