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.