In this dissertation we construct and study topological spaces that generalize spaces of quasi-invariants of finite reflection groups introduced in a recent work of Yu. Berest and A. C. Ramadoss. These spaces are obtained by applying the classical fibre-cofiber construction and its generalization to classical fibrations associated to compact connected Lie groups. Our results can be viewed as a natural extension of results of Berest and Ramadoss to higher rank Lie groups. We give a number of explicit examples and computations. These examples include the classifying spaces of classical Lie groups of arbitrary rank, associated homogeneous spaces as well as classifying spaces of commutativity of classicalLie groups introduced by A. Adem and J. Gómez. As an application, we compute the equivariant K-theory of the towers of homotopy fibers for the case of classifying spaces of classical Lie groups and compare our results to those of Berest and Ramadoss in the rank one case. Additionally, we explore spherical fibrations and consider conjugation action in the rank one case. We give explicit combinatorial presentations and study algebraic properties of rings of generalized quasi-invariants arising from the proposed topological construction.