The application of polymer fibers has rocketed to unimaginable heights in recent years and occupies every corner of our day-to-day life, from knitted protective textile clothes to buzzing smartphone electronics. Polymer fibers could be obtained from natural and synthetic polymers at a length scale from the nanometer to micrometer range. These fibers could be formed into different configurations such as single, core-sheath, hollow, blended, or composite according to human needs. Of these several conformations of fibers, core-sheath polymer fibers are an interesting class of materials, which shows superior physical, chemical, and biological properties. In core-sheath fiber structures, one of the components called a core is fully surrounded by the second component known as a sheath. In this format, different polymers can be applied as a sheath over a solid core of another polymer, thus resulting in a variety of modified properties while maintaining the major fiber property. After a brief introduction to core-sheath fibers, this review paper focuses on the development of the electrospinning process to manufacture core-sheath fibers followed by illustrating the current methodology and approaches to form them on a larger scale, suitable for industrial manufacturing and exploitation. Finally, the paper reviews the applications of the core-sheath fibers, in particular, recent studies of core-sheath polymer fibers in tissue engineering (nerve, vascular grafts, cardiomyocytes, bone, tendons, sutures, and wound healing), growth factors and other bioactive component release, and drug delivery. Therefore, core-sheath structures are a revolutionary development in the field of science and technology, becoming a backbone to many emerging technologies and novel opportunities.