This thesis is devoted to analyze the conductivity problem in the presense of an inclusion with multilayer structure. In the first part, we analyze the gradient blow-up of the solution to the conductivity problem in two dimensions in the presence of an inclusion with eccentric core-shell geometry. When the core and shell have circular boundaries that are nearly touching, the gradient of the solution can be arbitrary large in the narrow gap when the conductivities degenerate to zero or infinity. We derive an asymptotic formula for the solution in terms of the single and double layer potentials with image line charges. In the second part, we introduce the new concept of the geometric multipole expansion for the two-dimensional conductivity problem of which basis functions are associated with the inclusion's geometry. As an application, we construct multi-coated neutral inclusions of general smooth shape that have negligible perturbation for low-order polynomial loadings. We also suggest a new method to reconstruct the shape of inclusion.