Every people have their own unique fingerprints (epidermal ridges) and this gives the reason why the fingerprints could have been used as a means of personal identification for more than two thousand years. Since fingerprints do not change its shape in life, they have received substantial interests in the forensic science. There is no doubt that fingerprints have been widely studied scientifically in various fields. From the mathematical point of view, it is suggested that the fingerprint patterns are developed as the result of a buckling instability in the basal cell layer of the fetal epidermis. In order to analyze the general shape and pattern type of the fingerprints, it is essential to construct an appropriate mathematical model first. In the beginning of this thesis, we investigate the nonlinear version of the von Karman equations as a two dimensional mathematical modeling for the fingerprint formation. Then we see that the buckling process is governed by the compressive stress in the basal layer, not the curvature of the epidermal skin surface. The direction of epidermal ridges will be perpendicular to the direction of the greatest compressive stress. In the second half, based on the theory of elastostatics, we investigate partial differential equations as a three dimensional mathematical modeling for the fingerprint formation. We regard a finger-shaped shell as a special case of a bending plate with certain kinematic and mechanical assumptions. With the Reissner-Mindlin plate theory, we examine a framework for the adequate boundary-value problem.