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.