Let H be the upper half plane and F a discrete subgroup of AutH. When H mo d F is compact, one knows that the moduli space of unitary representations of F has an algebraic interpretation (cf. [7] and [10]); for example, if moreover F acts freely on H, the set of isomorphism classes of unitary representations of F can be identified with the set of equivalence classes of semi-stable vector bundles of degree zero on the smooth projective curve H modF, under a certain equivalence relation. The initial motivation for this work was to extend these considerations to the case when H m od F has finite measure. Suppose then that H modF has finite measure. Let X be the smooth projective curve containing H modF as an open subset and S the finite subset of X corresponding to parabolic and elliptic fixed points under F. Then to interpret algebraically the moduli of unitary representation of F, we find that the problem to be considered is the moduli of vector bundles V on X, endowed with additional structures, namely flags at the fibres of V at PeS. We call these quasi parabolic structures of V at S and, if in addition we attach some weights to these flags, we call the resulting structures parabolic structures on V at S (cf. Definition 1.5). The importance of attaching weights is that this allows us to define the notion of a parabolic degree (generalizing the usual notion of the degree of a vector bundle) and consequently the concept of parabolic semi-stable and stable vector bundles (generalizing Mumford's definition of semi-stable and stable vector bundles). With these definitions one gets a complete generalization of the results of [7, 10, 12] and in particular an algebraic interpretation of unitary representations of F via parabolic semi-stable vector bundles on X with parabolic structures at S (cf, Theorem 4.1). The basic outline of proof in this paper is exactly the same as in [12], however, we believe, that this work is not a routine generalization. There are some new aspects and the following are perhaps worth mentioning. One is of course the idea of parabolic structures; this was inspired by the work of Weil (cf. [16], p. 56). The second is a technical one but took some time to arrive at, namely when one