The author defines a particular kind of complex manifold with non-trivial topology, called annular bundle, which is essentially a holomorphic fiber bundle with fibers biholomorphic to plane annuli. It is possible to associate in a natural manner two invariants to an annular bundle - a function and a real (1,1)-form defined on the base of the bundle - called modular data, measuring the twisting of the complex structure along the fibers of the bundle. The first problem addressed in the paper is to study if (and how much) the modular data determine the annular bundle. It turns out that the answer depends on the topological structure of the base; in particular, it is affirmative if the homology groups in dimension 1 and 2 of the base vanish. Along the way, the author characterizes the real (1,1)-forms that may arise as curvature forms of a hermitian metric on a specific complex line bundle. The second half of the paper is devoted to the study of fiber maps between annular bundles. It is possible to define a concept of harmonic measure on an annular bundle, fully generalizing the classical theory of Landau and Osserman. The interplay between the harmonic measure and the modular data allows to describe completely the automorphism group of an annular bundle, and to prove a rigidity theorem for fiber self-maps on a large class of annular bundles.