Electrospinning and wet-spinning have been recognized as two of the most efficient and promising techniques for producing polymeric fibrous constructs for a wide range of applications, including optics, electronics, food industry and biomedical applications. They have gained considerable attention in the past few decades because of their unique features and tunable architectures that can mimic desirable biological features, responding more effectively to local demands. In this review, various fiber architectures and configurations, varying from monolayer and core-shell fibers to tri-axial, porous, multilayer, side-by-side and helical fibers, are discussed, highlighting the influence of processing parameters in the final constructs. Additionally, the envisaged biomedical purposes for the examined fiber architectures, mainly focused on drug delivery and tissue engineering applications, are explored at great length.