The goal of this publication is to provide basic tools of differential topology to study systems of nonlinear equations, and to apply them to the analysis of general equilibrium models with complete and incomplete markets. The main content of general equilibrium analysis is to study existence, (local) uniqueness and efficiency of equilibria. To study existence Differential Topology and General Equilibrium with Complete and Incomplete Markets combines two features. As a first step, first order conditions (of agents' maximization problems) and market clearing conditions, instead of aggregate excess demand functions, are used. As a second step, a homotopy argument, stated and proved in relatively elementary manner, is applied to that "extended systemof equations. Local uniqueness and smooth dependence of the endogenous variables from the exogenous ones are studied using a version of a so called parametric transversality theorem. In a standard general equilibrium model, all equilibria are efficient, but that is not the case if some imperfection, like incomplete markets, asymmetric information, strategic interaction, is added. Then, for almost all economies, equilibria are inefficient, and an outside institution can Pareto improve upon the market outcome. Those results are proved showing that a well-chosen system of equations has no solutions