We study the extent to which the Galvin-Mycielski-Solovay characterization of strong measure zero subsets of the real line extends to arbitrary Polish groups. In particular, we show that an abelian Polish group satisfies the GMS characterization if and only if it is locally compact. We shall also consider the non-abelian case, and discuss the use and existence of invariant submeasures on Polish groups

(joint with W. Wohofsky, J. Zapletal and/or O. Zindulka).