  In this paper, we study the perfect and the insulated conductivity problems
with multiple inclusions imbedded in a bounded domain in $\mathbb{R}^n, n\ge
2$. For these two extreme cases of the conductivity problems, the gradients of
their solutions may blow up as two inclusions approach each other. We establish
the gradient estimates for the perfect conductivity problems and an upper bound
of the gradients for the insulated conductivity problems in terms of the
distances between any two closely spaced inclusions.
