Inverse boundary value problems originated in early 80’s, from the contribution of A.P. Calderon on the inverse conductivity problem [C], in which one attempts to recover the electrical conductivity of a body by means of boundary measurements on the voltage and current. Since then, the area of inverse boundary value problems for linear elliptic equations has undergone a great deal of development [U]. The theoretical growth of this area contributes to many areas of applications ranging from medical imaging to various detection techniques [B-B][Che-Is]. In this paper we discuss several conjectures in the inverse boundary value problems for quasilinear elliptic equations and their recent progress. These problems concern anisotropic quasilinear elliptic equations in connection with nonlinear materials and the nonlinear elasticity system.