Here, we review the state-of-the-art in the field of engineered self-healing materials. These materials mimic the functionalities of various natural materials found in the human body (e.g., the healing of skin and bones by the vascular system). The fabrication methods used to produce these "vascular-system-like" engineered self-healing materials, such as electrospinning (including co-electrospinning and emulsion spinning) and solution blowing (including coaxial solution blowing and emulsion blowing) are discussed in detail. Further, a few other approaches involving the use of hollow fibers are also described. In addition, various currently used healing materials/agents, such as dicyclopentadiene and Grubbs' catalyst, poly(dimethyl siloxane), and bisphenol-A-based epoxy, are described. We also review the characterization methods employed to verify the physical and chemical aspects of self-healing, that is, the methods used to confirm that the healing agent has been released and that it has resulted in healing, as well as the morphological changes induced in the damaged material by the healing agent. These characterization methods include different visualization and spectroscopy techniques and thermal analysis methods. Special attention is paid to the characterization of the mechanical consequences of self-healing. The effects of self-healing on the mechanical properties such as stiffness and adhesion of the damaged material are evaluated using the tensile test, double cantilever beam test, plane strip test, bending test, and adhesion test (e.g., blister test). Finally, the future direction of the development of these systems is discussed.