Which rings can arise as cohomology rings of algebraic varieties?
To be more specific, take a Weil cohomology theory $H^*$ with coefficients in a field $K$ of characteristic 0 defined for smooth projective varieties over an algebraically closed field $k$.
Are there any conjectures and results giving criteria for a finite-dimensional graded-commutative $K$-algebra to be realizable as $H^*(V_k, K)$ for a smooth projective variety $V_k$ over $k$? Fixing such a $K$-algebra $A_K$ what can one say about the set of $V_k$ with $H^*(V_k, K)=A_K$?
I suppose the title question makes sense in purely topological settings too, for any cohomology theory with ring structure.