A pen-nutation group is said to be quasiprimitive if every nontrivial normal subgroup is transitive. Every primitive permutation group is quasiprimitive, but the converse is not true. Quasiprimitive group actions arise naturally in the investigation of many combinatorial structures, such as arc-transitive graphs, and line-transitive finite geometries. We describe some of the properties shared by finite primitive and quasiprimitive permutation groups, and some of their differences. We also indicate some of the recent major combinatorial applications of finite quasiprimitive groups. (C) 2002 Elsevier Science B.V. All rights reserved.