No fixed age. It is hard to fix on a number because there are broad differences between two individuals. It is better to categorize the physiological age rather than the chronological age of a patient. The question to really ask is "does the risk of transplanting a solid organ outweigh the benefit of receiving that organ?" the answer can be subjective and viewed differently among professionals.
Rare after 75. Based on US Renal Data System registry (www.usrds.org) only 0.4% of 53,000 patients starting dialysis received a kidney transplant. For age 65-74 3.8% of 50,000 starting dialysis received a transplant. Most of the latter group were under 70. Age related medical problems especially cardiovascular disease, cancer, or infections are the contraindications for transplant rather than age itself.
Usually 70-75. At least for a liver transplant, usually 70-75. More importantly is the absence of other medical conditions. So, if a patient has liver failure without diabetes, hypertension, heart disease, or kidney disease, programs are more likely to consider them for a transplant. I believe this also generally true for other organs as well.