Stretchable displays, capable of freely transforming their shapes, have received significant attention as alternatives to conventional rigid displays, and they are anticipated to provide new opportunities in various human-friendly electronics applications. As a core component of stretchable displays, high-performance stretchable light-emitting diodes (LEDs) have recently emerged. The approaches to fabricate stretchable LEDs are broadly categorized into two groups, namely "structural" and "material-based" approaches, based on the mechanisms to tolerate strain. While structural approaches rely on specially designed geometries to dissipate applied strain, material-based approaches mainly focus on replacing conventional rigid components of LEDs to soft and stretchable materials. Here, we review the latest studies on the fabrication of stretchable LEDs, which is accomplished through these distinctive strategies. First, we introduce representative device designs for efficient strain distribution, encompassing island-bridge structures, wavy buckling, and kirigami-/origami-based structures. For the material-based approaches, we discuss the latest studies for intrinsically stretchable (is-) electronic/optoelectronic materials, including the formation of conductive nanocomposite and polymeric blending with various additives. The review also provides examples of is-LEDs, focusing on their luminous performance and stretchability. We conclude this review with a brief outlook on future technologies.