I bought a memory foam mattress several years ago due, in part, to lower back problems. After sleeping on it for years and receiving no back relief, I finally just swapped it for the old mattress in my guest room. They have their pros and cons. As Aunty mentioned, they are definitely much warmer than normal mattresses. You can buy special liners that will help but it is still warmer than normal.

I would say that they generally make for nice mattresses but they aren't worth all the hype they've generated.

With regards to the box-spring question, I don't think it matters what you have under the memory foam. Mine is really thick and I doubt it would make a difference if there was a concrete slab or a box spring under it.