The quest to imbue machines with intelligence akin to that of humans, through the development of adaptable neuromorphic devices and the creation of artificial neural systems, has long stood as a pivotal goal in both scientific inquiry and industrial advancement. Recent advancements in flexible neuromorphic electronics primarily rely on nanomaterials and polymers owing to their inherent uniformity, superior mechanical and electrical capabilities, and versatile functionalities. However, this field is still in its nascent stage, necessitating continuous efforts in materials innovation and device/system design. Therefore, it is imperative to conduct an extensive and comprehensive analysis to summarize current progress. This review highlights the advancements and applications of flexible neuromorphics, involving inorganic nanomaterials (zero-/one-/two-dimensional, and heterostructure), carbon-based nanomaterials such as carbon nanotubes (CNTs) and graphene, and polymers. Additionally, a comprehensive comparison and summary of the structural compositions, design strategies, key performance, and significant applications of these devices are provided. Furthermore, the challenges and future directions pertaining to materials/devices/systems associated with flexible neuromorphics are also addressed. The aim of this review is to shed light on the rapidly growing field of flexible neuromorphics, attract experts from diverse disciplines (e.g., electronics, materials science, neurobiology), and foster further innovation for its accelerated development.