if g is a group acting on a vector space v by linear transformations, then the invariant polynomial functions on v form a ring. in this talk we will discuss upper bounds for the degrees of generators of this invariant ring. an example of particular interest is the action of the group $sl_n x sl_n$ on the space of m-tuples of n x n matrices by simultaneous left-right multiplication. in this case, visu makam and the speaker recently proved that invariants of degree at most $mn^4$ generate the invariant ring. we will explore an interesting connection between this result and the notion of noncommutative rank.