Are there any general methods for computing fundamental group or singular cohomology of a projective variety (over C of course), if given the equations defining the variety?

I seem to recall that, if the variety is smooth, we can compute the H^{p,q}'s by computer -- and thus the H^n's by Hodge decomposition -- is this correct? However this won't work if the variety is not smooth -- are there any techniques that work even for non-smooth things?