The amniotic membrane (AM) is the innermost part of the fetal placenta, which surrounds and protects the fetus. Due to its structural components (stem cells, growth factors, and proteins), AMs display unique biological properties and are a widely available and cost-effective tissue. As a result, AMs have been used for a century as a natural biocompatible dressing for healing corneal and skin wounds. To further increase its properties and expand its applications, advanced hybrid materials based on AMs have recently been developed. One existing approach is to combine the AM with a secondary material to create composite membranes. This review highlights the increasing development of new multilayer composite-based AMs in recent years and focuses on the benefits of additive manufacturing technologies and electrospinning, the most commonly used strategy, in expanding their use for tissue engineering and clinical applications. The use of AMs and multilayer composite-based AMs in the context of nerve regeneration is particularly emphasized and other tissue engineering applications are also discussed. This review highlights that these electrospun multilayered composite membranes were mainly created using decellularized or de-epithelialized AMs, with both synthetic and natural polymers used as secondary materials. Finally, some suggestions are provided to further enhance the biological and mechanical properties of these composite membranes.