We introduce and study a significant generaliza­tion of Abel's summability method, and their corresponding limiting process. This leads to an analogue to Hardy-Littlewood Tauberian Theorem.The first section includes an introduction to some basic concepts of summability methods and a survey of classical and neoclassical results. In the second section a general summability method is designed and some related Tauberian theorems are established. In the third section higher order of Abel's summability methods are obtained as a special case of a general summability method and the general Littlewood theorem is proved for those summabil­ity methods. Finally we give Tauberian theorems corresponding to $(C, m)$-summability methods and present some further convergence theorems.