If $V$ is a representation of a linear algebraic group $G$, a set $S$ of $G$-invariant regular functions on $V$ is called separating if the following holds: If two elements $v,v'$ from $V$ can be separated by an invariant function, then there is an f from S such that $f(v)$ is different from $f(v')$. It is known that there always exist finite separating sets, even though the invariant ring might not be finitely generated. Moreover, if the group $G$ is finite, then the invariant functions of degree $\le |G|$ always form a separating set. So the degree bounds are definitely smaller than for the generators of the invariants. Jointly with Martin Kohls we have shown that for a non-finite linear algebraic group G such an upper bound for the degrees of a separating set does not exist. Moreover, for a finite group G we define b(G) to be the minimal number d such that for every G-module V there is a separating set of degree less or equal to d. We then show that for a subgroup H of G we have $b(H) \le b(G) \le [G:H] b(H)$, and that $b(G) \le b(G/H) b(H)$ in case H is normal. In addition, we calculate $b(G)$ for some specific finite groups.