Much of the current research on embryonic stem cells has been done on mice. In mice, researchers have the advantage of being able to use syngenic stem cells — that is, embryonic stem cells with the same genetic makeup as the host — due to the extensive selective breeding that has been done on laboratory mice.
With syngenic embryonic stem cells, researchers have had great difficulty causing the stem cells to differentiate properly. For example (Nussbaum 2007), in experimental heart disease treatment, the goal of the research is to cause the stem cells to become heart muscle cells, but doing this in vivo (that is, inside the body) has been challenging. Instead, the stem cells form masses of undifferentiated cells. These non-cancerous tumors are called teratomas. Even when placed in injured heart tissue, which hypothetically should give chemical cues to the stem cells to become heart muscle cells, the stem cells instead form teratomas. Research continues on how to solve this significant problem.
With allogenic embryonic stem cells, the teratomas are eliminated by the immune system and therefore do not cause problems. Unfortunately, they also do not help treat the disease they are intended to treat.