We consider elusive permutation groups. Many examples of such groups are introduced, including the Mathieu group $M_{11}$ as an important example. We describe the known constructions of elusive groups and discuss the possible degrees of elusive groups. We classify all primitive elusive groups. We introduce the polycirculant conjecture and show that none of the described constructions of elusive groups leads to a counterexample for this conjecture. We also show that no transitive group of automorphisms of a vertex transitive graph of valency $3$ or $4$ is elusive. We also present some results on the highest order semiregular automorphism of a cubic vertex transitive graph.
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