A transitive permutation group is called semiprimitive if every normal subgroup is 
either transitive or semiregular. This class of groups includes all primitive, 
quasiprimitive and innately transitive groups. They have received recent attention 
due to a conjecture of Poto\v{c}nik, Spiga and Verret that generalises the Weiss conjecture 
for locally primitive graphs. In this talk I will discuss recent work with Luke Morgan 
that develops a general theory  for the structure of semiprimitive groups and give 
some applications of the new theory.