The Weiss Conjecture has stimulated much study of symmetric graphs, especially concerning the order of vertex-stabilisers. The conjecture roughly asserts that in a locally-primitive symmetric graph, the vertex-stabiliser is not too large compared to the valency. The conjecture implies that “local properties” have a global impact, since this also bounds the order of the automorphism group. Further refinements to this conjecture were suggested, first by Praeger who asked if local quasiprimitivity might be sufficient, and then by Potocnik-Spiga-Verret who found semiprimitive groups are the key. This latter class of permutation groups has not been well studied, and indeed only recently – 2008 – appeared in the literature. In seeking to understand the validity of these conjectures, I have been motivated to study semiprimitive groups in a more general context, and try to make our understanding of this class of groups equal to that of the well studied class of primitive groups. In this talk I’ll mention some progress towards that, and highlight the impact the results have already had on the aforementioned conjectures.