Endoscopy structures of automorphic forms was one of the
basic structures discovered through the Arthur-Selberg trace formula method to establish the Langlands functoriality for classical groups.

In this talk, we will discuss my recent work on characterization of the endoscopy structure in terms of the order of pole at s=1 of certain L-functions, and in terms of a family of periods of automorphic forms, which was discovered jointly with David Ginzburg. At the end, I may discuss how to contruct the
endoscopy transfer by integral operators, which is
a joint work with Ginzburg and Soudry.

We shall discuss some results which classify the smallest degree representations of simple groups. We will then discuss various applications of these ideas including the solution of a conjecture of Serre on semisimplicity of certain representations and a conjecture of Kollar and Larsen on the non-irreducibility of symmetric powers.