  It has long been known that there exist finite connected tetravalent
arc-transitive graphs with arbitrarily large vertex-stabilisers. However,
beside a well known family of exceptional graphs, related to the lexicographic
product of a cycle with an edgeless graph on two vertices, only a few such
infinite families of graphs are known. In this paper, we present two more
families of tetravalent arc-transitive graphs with large vertex-stabilisers,
each significant for its own reason.
