ABSTRACT: In this talkwediscuss the applications of finite permutationgroups to combinatorial designs. We will divide this talk into the following seven sections:

(1) Definition of 2 − (v, k , λ) designs and examples; (2) Automorphism groups of designs, elementary properties; (3) The socle of Aut(D), where D is a 2 − (v, k , 1) design; (4) The studies on block transitive 2 − (v, k , 1) designs with k small, (5) Some works on designs with block transitive and solvable automorphism groups; (6) Simple groups of Lie type of low rank act on designs; (7) Classical groups of high dimensions act on designs.