My answer, originally published on Quora.
Q: What illnesses may disqualify someone from adopting a child internationally?
If one of parents has a history of cancer and doctors said < 1% of its coming back, may the family apply to adopt a child?
This varies according to the country you are adopting from, and might also vary by the adoption agencies you use in your home country and abroad.
As a data point, in 2007 (when we adopted), you'd be completely ineligible to adopt from South Korea if you were obese (had a BMI above 35) or were taking medication for depression. For other noteworthy or chronic medical conditions, Korea would consider them on a case-by-case basis. These included things like a history of cancer, a history of depression (even if not currently on meds) or other mental illness, a positive TB skin test, and being overweight (BMI between 30 and 35).
In general, for most agencies and most countries, you'll be asked to disclose any and all medical conditions, past and present. They will likely also require your doctor to fill out and certify a detailed questionnaire about your present health and past medical history, including listing all medications and past surgeries. You will probably need to submit test results for conditions like HIV, Hep C, and TB. They might also ask for your medical records. For anything out of the ordinary, you'll have to explain it and why it won't affect your ability to parent.
I'd encourage you to call up several adoption agencies that do a lot of adoptions in the country you are considering. Be frank with them and ask their honest assessment of your chances of getting approved. You don't want to go through the large effort and expense of applying, just to be rejected. I'd ask several agencies, not just one, because there are agencies that will gladly take your money to start the process even if they secretly know it's unlikely you'll get approved. I'd also find some online forums or email lists for waiting adoptive parents for your preferred country, and ask if anyone in similar circumstances has been matched with a child recently.
If you get the okay and start the application process, I would:
Write a letter explaining why your history of cancer will not affect ability to a child a loving home and raise him/her to adulthood.
Ask your doctor to write a similar letter.
Have both letters translated into your target county's language by a certified translater with lots of experience in medical translation. (You don't want your agency's billingual social workers, who are not doctors or translation professionals, to mess up the translation. Hire your own professionals.