  The relationship between comodules of a coring and flat connections is
reviewed. In particular we specialise to corings which are built on a tensor
product of algebra and a coalgebra. Such corings are in one-to-one
correspondence with entwining structures, and their comodules are entwined
modules. These include Yetter-Drinfeld and anti-Yetter-Drinfeld modules and
their generalisations, hence all the modules of interest to Hopf-cyclic
cohomology. In this way the interpretation of the latter as modules with flat
connections [A Kaygun and M Khalkhali, Hopf modules and noncommutative
differential geometry, Lett. Math. Phys. 76 (2006), 77--91] is obtained as a
corollary of a more general theory. We also introduce the notions of a
connection in a comodule and of a bicomodule connection, and show how comodules
with flat connections can be interpreted as modules of a C-ring. In this way
all the above mentioned Hopf modules can be interpreted as comodules with flat
connections.
