Over the course of the last century, organ transplantation has overcome major technical limitations to become the success it is today. The breakthroughs include developing techniques for vascular anastomoses, managing the immune response (initially by avoiding it with the use of identical twins and subsequently controlling it with chemical immunosuppressants), and devising preservation solutions that enable prolonged periods of ex vivo storage while preserving function. One challenge that has remained from the outset is to overcome the shortage of suitable donor organs. The results of organ transplantation continue to improve, both as a consequence of the above innovations and the improvements in peri- and postoperative management. This review describes some of the achievements and challenges of organ transplantation.