I don't think centaurs would exist without magical or divine intervention, except in myth and dreams, for a number of reasons other answers have pointed out, such as no evolutionary path, and the fact you're combining two mostly-complete bodies, but one lacks a place to excrete (not SO bad - probably it goes into the horse-body digestive system...) and the other lacks the head (so the humanoid head would need to be able to run the horse body.
I think it might in theory be possible as the mad project of some crazy god or mad scientist or something, but it would require some sort of grafting with futuristic anti-rejection technology, incredible plastic surgery, and/or a lot of medical and/or genetic handwavium technology.
So the issues include:
- immune system rejection of flesh and blood
- horse brain functions
- coordination of the two bodies
- nervous system and how the spinal column works
What makes me think it might theoretically be possible to engineer (hey, it's more productive than figuring out how to blow up a star or make a Dyson Sphere or go power-mad at everyone else's expense), is the example of some conjoined (?) twins where there is a shared body sometimes with extra parts, and two heads.
One of my more awful ideas is there might need to be a horse brain somewhere around the inside of the lower humanoid torso... maybe not.
Maybe there's a way to screw with the genetics of horses and humans enough to make one set of chromosomes that will grow a horse body and a human torso and have them attached the way the Greeks imagined centaurs. I really doubt it, but I wouldn't utterly rule it out. Maybe it just has one brain that's someplace between human and equestrian, and that somehow figures out how everything's connected - I would not have thought some conjoined twin examples would survive, which did.
So my theoretical centaur might have a hybrid brain that handles everything, in the humanoid skull. The DNA is some engineered weird hybrid that reliably grows a functional, viable, aesthetically pleasing centaur. I imagine the growth rate would be pretty weird, since horses grow up faster than humans... there might be some large horse bodies with childish human torsos attached unless more effort is made to somehow accelerate the human growth, though the human brain takes time to learn things, so I imagine the psychology of a centaur would be horse-dominated especially before it grew up a lot, which might be about time the horse body would tend to be wearing out, unless somehow that tendency was overcome. The digestive system needs to either be horselike or human or ideally both with the ability to route appropriately - good luck with that. The nervous system should connect all to the one brain. The respiratory system is probably a huge bitch to figure out, maybe, since there's one mouth and it needs to get air for the horse body too... um, I suggest a BLOWHOLE on the back, for the horse lungs. Probably two hearts, like some human twins, and thus two circulatory systems, but hopefully compatible blood... hmm.
An alternative might be to have the humanoid organs partly or wholly un-needed, un-used, or even absent, just somehow shaped like a human (perhaps keep the lungs), but be mainly just muscle shaped more like a human torso than a horse neck, but for non-muscular purposes function more like a horse neck. So mainly you've got a horse with extra human arms and shoulders and head. Of course horses like to eat lots of hay, and human teeth aren't great with hay.
In any case, even if it is possible, I'd say it's the kind of project that would only be undertaken by people with massive resources and nothing they think is better to do with those resources.
Though maybe in the distant future there will be genetic breakthroughs that are useful for other things, which end up making centaurs not that big a deal. Certainly there is sci fi which postulates that kind of clean easy happy outcomes of genetic screwery.