Federated learning (FL) and bioinspired computing (BIC), two distinct, yet complementary fields, have gained significant attention in the machine learning community due to their unique characteristics. FL enables decentralized machine learning by allowing models to be trained on data residing across multiple devices or servers without exchanging raw data, thus enhancing privacy and reducing communication overhead. Conversely, BIC draws inspiration from nature to develop robust and adaptive computational solutions for complex problems. This paper explores the state of the art in the integration of FL and BIC, introducing BIC techniques and discussing the motivations for their integration with FL. The convergence of these fields can lead to improved model accuracy, enhanced privacy, energy efficiency, and reduced communication overhead. This synergy addresses inherent challenges in FL, such as data heterogeneity and limited computational resources, and opens up new avenues for developing more efficient and autonomous learning systems. The integration of FL and BIC holds promise for various application domains, including healthcare, finance, and smart cities, where privacy-preserving and efficient computation is paramount. This survey provides a systematic review of the current research landscape, identifies key challenges and opportunities, and suggests future directions for the successful integration of FL and BIC.