Use of endovascular hypothermia in organ and/or tissue transplantations

Methods for (a) preventing hypoxic damage to a potentially transplantable organ or tissue prior to explanation of that organ or tissue from the body of a mammalian transplant donor and (b) preventing rejection of a transplanted organ or tissue in a human or veterinary transplant recipient. The methods comprise placing a heat exchange apparatus in the vasculature of the donor or recipient and using that heat exchange apparatus to cool at least a portion of the body of the donor or recipient to a temperature below normothermia (e.g. below normothermia and sometimes between about 30° C. and about 36° C.).