Investigate on the Criterion of Uniform Convergence Mathematics and Applied Mathematics 2006-2 Jiang Su-ping Supervisor Liang Zhi-qing Abstract Uniform Convergence theory is an important research branch of mathematical analysis. The understanding and judging of this conception are the key as well as difficult point of mathematical analysis. Further more, Uniform Convergence has been widely used in the subjects of Functional Analysis and Partial Differential Equations. This article will first briefly explain the Function Column, Series of Functions and Parameter Improper concept of uniform convergence. Then, out from three aspects, namely the function, the function parameters of the Series and the infinite integration with parameter, it will list some methods commonly used in the identification of Uniform Convergence from which some theorem will be deduced. In the research of the methods of identifying Uniform Convergence, another kind of identifying method called Ratio method is deduced through between discriminant method. Besides, taking advantage of L condition, this paper will define Uniform L condition and discusses Convergence under L condition. Besides, it will discusse the Uniform Convergence of function when its derived functions are uniformly bounded under micro-conditions. In the research of the methods of identifying Uniform Convergence of Series, this paper will give the definition of L condition of Uniform Convergence of Series and discusses Uniform Convergence of Series under L condition. Theorems that has not been proved in document 2 will also be briefly proved in this paper. Key words: function column; series of functions; infinite integration with parameter; uniform convergence