This is an expository thesis, which reviews the recent work of Agnew, Vuckovic, Martic and Jakimovski on the Lototsky summability method and its generalisations. An introductory sketch of summability theory is provided, and it is noted that a more general form of the Lototsky method was already introduced by Karamata In 1935. In reviewing the recent work some minor errors are corrected, and details of proofs, or in some cases entire proofs, are supplied. In particular, the theorem stated by Martic, that the Karamata method includes the Euler, is proved in detail. Part of the work is clarified by the use of finite difference theory (Stirling's numbers), and some other simplifications are made.