As has already been stated the main problem with Substance Dualism is the problem first brought up by Princess Elisabeth of Bohemia: If mind and body are two separate entities then how do they interact with each other?
Descartes' response, and I'm paraphrasing, was 'The soul exists in another realm to the physical. It doesn't work in the same ways as the physical and is really, really, really, really, really complicated.'
To which Elisabeth of Bohemia replied (again: paraphrasing) "Sounds like a cop-out answer to me."
As much as I agree with Elisabeth of Bohemia, I think her objection isn't really strong enough to invalidate the end result of all six of Descartes' meditations. The best, or at least most interesting, argument I have seen against substance dualism comes from Jaegwon Kim who makes an argument that goes something like:
"Say we have two rifles, 'A' and 'B' and we have two apples 'a' and 'b'. Now say rifle 'A' is shot at apple 'a' and apple a explodes. Now say rifle 'B' is shot at apple 'b' and apple b explodes. How can we say that it was rifle A that causes apple a's explosion and not rifle B? Well, we simply refer to the rifle's position in space. It must have been rifle A that caused apple a's explosion because rifle A was in such a position that it was pointed toward apple a when it was fired.
Now, say we have two immaterial minds, 'A' and 'B' and we have two material bodies 'a' and 'b'. A feels like getting some chocolate, so a leaves its house and goes to the store to buy some. B feels like getting some chocolate, so b leaves its house and goes to the store to buy some. How can we say that it was A which caused a's movement and not B? Remembering that A and B are immaterial and take up no space."
There are obvious problems with Jaegwon Kim's arguments, but I do find this argument against substance dualism to be interesting.