Human umbilical cord (UC) has been a tissue of increasing interest in recent years. Many groups have shown the stem cell potency of stromal cells isolated from the human UC mesenchymal tissue, namely, Wharton's jelly. Since UC is a postnatal organ discarded after birth, the collection of cells does not require an invasive procedure with ethical concerns. Stromal cells, as the dominant cells of this fetus-derived tissue, possess multipotent properties between embryonic stem cells and adult stem cells. They bear a relatively higher proliferation rate and self-renewal capacity. Although they share common surface markers with bone marrow-derived MSCs, they also express certain embryonic stem cell markers, albeit in low levels. Without any spontaneous differentiation, they can be successfully differentiated into mature adipocytes, osteoblasts, chondrocytes, skeletal myocytes, cardiomyocytes, neurons, and endothelial cells. While causing no immunorejection reaction, they effectively function in vivo as dopaminergic neurons, myocytes, and endothelial cells. Given these characteristics, particularly the plasticity and developmental flexibility, UC stromal cells are now considered an alternative source of stem cells and deserve to be examined in long-term clinical trials. This review first aims to document the published findings so far regarding the nature of human UC stroma with special emphasis on the spatial distribution and functional structure of stromal cells and matrix, which serves as a niche for residing cells, and, secondly, to assess the in vitro and in vivo experiments in which differential stem cell potencies were evaluated.