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Two basic approaches to device modeling are presented. The physical approach consists of 4 basic steps: 1) device physics analysis and partitioning, 2) physical equation formulation, 3) equation simplification and solution, and 4) nonlinear network synthesis. The black-box approach consists also of 4 basic steps: 1) experimental observations, 2) mathematical modeling, 3) model validation, and 4) nonlinear network synthesis. Each approach is ilustrated with 2 examples: Gunn Diode and SCR for the physical approach and Hysteretic Inductor and Memristive Device for the black-box approach. While the techniques for carrying out the first 3 steps in each approach presently involve more art than science, a unified theory for carrying out the last step (nonlinear network synthesis) is beginning to emerge. In particular, the universe of all lumped nonlinear circuit elements can now be classified into algebraic and dynamic elements via a completely logical axiomatic approach. Contrary to what is the case in linear circuit theory, it is shown that an infinite variety of basic algebraic and dynamic elements will be needed in the eventual formulation of a unified theory on device modeling. Consequently, these elements are given a complete and in-depth treatment in this paper. This material can also be regarded as a self-contained survey of the state-of-the-art on nonlinear network synthesis.