This paper presents the planar approximation to field theory through the limit of a large internal symmetry group. The authors investigate the planar diagrams in field theory and show that this approach provides a powerful method for counting planar diagrams. Results are given for cubic and quartic vertices, some of which are new. The paper also shows that quantum mechanics in this approximation is equivalent to a free Fermi gas system. The paper begins with an introduction, discussing the motivation for studying the planar approximation to field theory. The authors hope that this approach may provide a way to perform reliable computations in the large coupling phase of non-abelian gauge fields in four dimensions. They also note that such topological expansions may be related to dual string models. The paper also discusses the simplifications that occur in the large N-limit for linear or non-linear σ-models, which allow for the discrimination of symmetry phases. A first part of the paper is devoted to preliminary combinatorial aspects. The method used for this "zero-dimensional" field theory, in which every propagator is set equal to unity, is not combinatorial and may be extended to genuine calculations of Green functions in a real field theory. This enabled the authors to solve some counting problems whose solutions are not known. In Section 5, the authors compute explicitly the contribution of all planar Feynman diagrams to the ground state energy of a one-dimensional g x^4 anharmonic oscillator. The solution may be generalized to include the first non-planar corrections. Surprisingly, the problem can be restated as finding the ground state energy of a one-dimensional uninteracting Fermi gas, which is trivial. Finally, the authors note that in contrast with the true theory, the planar sum is analytic near the origin in the complex coupling constant space, revealing that the large field region of the Feynman path integral has been drastically modified.This paper presents the planar approximation to field theory through the limit of a large internal symmetry group. The authors investigate the planar diagrams in field theory and show that this approach provides a powerful method for counting planar diagrams. Results are given for cubic and quartic vertices, some of which are new. The paper also shows that quantum mechanics in this approximation is equivalent to a free Fermi gas system. The paper begins with an introduction, discussing the motivation for studying the planar approximation to field theory. The authors hope that this approach may provide a way to perform reliable computations in the large coupling phase of non-abelian gauge fields in four dimensions. They also note that such topological expansions may be related to dual string models. The paper also discusses the simplifications that occur in the large N-limit for linear or non-linear σ-models, which allow for the discrimination of symmetry phases. A first part of the paper is devoted to preliminary combinatorial aspects. The method used for this "zero-dimensional" field theory, in which every propagator is set equal to unity, is not combinatorial and may be extended to genuine calculations of Green functions in a real field theory. This enabled the authors to solve some counting problems whose solutions are not known. In Section 5, the authors compute explicitly the contribution of all planar Feynman diagrams to the ground state energy of a one-dimensional g x^4 anharmonic oscillator. The solution may be generalized to include the first non-planar corrections. Surprisingly, the problem can be restated as finding the ground state energy of a one-dimensional uninteracting Fermi gas, which is trivial. Finally, the authors note that in contrast with the true theory, the planar sum is analytic near the origin in the complex coupling constant space, revealing that the large field region of the Feynman path integral has been drastically modified.