Biomimicry applies the fundamental principles of natural materials, processes, and structures to technological applications. This review presents the two strategies of biomimicry - bottom-up and top-down approaches, using biomimetic polymer fibers and suitable spinning techniques as examples. The bottom-up biomimicry approach helps to acquire fundamental knowledge on biological systems, which can then be leveraged for technological advancements. Within this context, we discuss the biomimetic spinning of silk and collagen fibers due to their unique natural mechanical properties. To achieve successful biomimicry, it is imperative to carefully adjust the spinning solution and processing parameters. On the other hand, top-down biomimicry aims to solve technological problems by seeking solutions from natural role models. This approach will be illustrated using examples such as spider webs, animal hair, and tissue structures. To contextualize biomimicking approaches in practical applications, this review will summarize the use of biomimicry in filter technologies, textiles, and tissue engineering.