We consider countably additive, nonnegative, extended real-valued measures which vanish on singletons. Such a measure is universal on a set $X$ iff it is defined on all subsets of $X$ and is semiregular iff every set of positive measure contains a subset of positive finite measure. We study the problem of existence of a universal semiregular measure on $X$ which is invariant under a given group of bijections of $X$. Moreover we discuss some properties of universal, semiregular, invariant measures on groups.