The quantitative analysis of localized flux distributions in transformer cores has become easy through the progress of numerical field calculations. In this paper, the effects of core constructions, joint configurations, magnetizing characteristics of materials on the flux distributions are examined using newly developed techniques such as a gap element, an approximate method for solving three-dimensional magnetic fields, the time periodicity finite element method, an efficient technique for treating hysteresis characteristics and so on. A method for the optimum design of transformer cores is also discussed. Main results obtained can be summarized as follows: (a) The building factor of the core made of higely-oriented silicon steel is larger than that of the conventional core. (b) The iron losses at joints are much affected by overlap lengths, number of laminations per stagger layer and a small irregularity of the arrangement of sheets. (c) It is clarified that examinations of the optimum construction of core and the most desirable magnetic characteristics of core material are possible using the finite element method.