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In this paper we present a numerical shape optimization method of continua for solving min-max problems and identification problems. The min-max shape optimization problems involve minimization of maximum stress or maximum displacement; the shape identification problems involve the determination of shapes that achieve a given desired stress distribution or displacement distribution. Each problem is formulated and sensitivity functions are derived using the Lagrangian multiplier method and the material derivative method. The traction method, which is a shape optimization method, is employed to find the optimal domain variation that reduces the objective functional. The proposed numerical analysis method makes it possible to design optimal structures for maximizing strength and rigidity and for controlling stress and displacement distributions. Examples of computed results are presented to show the validity and practical utility of the proposed method.