This book exclusively deals with the study of ideal convergence on different type of sequences spaces. The notion of “ideal convergence” is the generalization of the “statistical convergence”, introduced by H. Fast, which is an extension of the usual concept of sequential limits. The book also discusses the applications of these non-matrix methods in approximation theory. Written in a self-contained style, the book discusses in detail the methods of ideal convergence for single sequences along with applications and suitable examples. The last chapter gives some applications in approximation theory; the results are expected to find application in many other areas of pure and applied mathematics such as mathematical analysis, probability, fixed point theory and statistics.