The computational theory of mind is the leading theory for pure cognition. There is no leading theory of consciousness, although some people also favor the computational theory of mind for that as well but this is far less popular.
I think that among those who believe in materialism, the only two common positions are either that consciousness is an emergent property of the computational mind / an illusion (I view these as somewhat similar, though you can disagree), or that consciousness is some kind of physical interaction that we have basically no understanding of; and of course there are plenty of "agnostics" on this topic. Among these, I think it's fair to say that the first group is the larger one among materialists.