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From the first time the author encountered General Linear Methods in general and Almost-Runge-Kutta methods in particular, and realized the enormous potentials and possibilities they provide, there has been a deep desire in the author, to draw the attention of researchers and mathematicians to these methods. As such, in this book the author has engaged in a detailed and comprehensive analysis of Almost Runge-Kutta (ARK) methods, derived new and effective methods especially for non-stiff differential equations. The book provides a comprehensive introduction to numerical methods for solving Ordinary Differential equations, it includes a detailed coverage of Runge-Kutta, linear multistep, and general linear methods,and thoroughly breaks down the derivation process of ARK methods. Extensive numerical experiments are carried out to see how the new ARK methods compare with some selected traditional methods and the results confirms the effectiveness and viability of ARK methods as a means by which Scientists, Mathematicians and Engineers can obtain accurate and reliable results for non-stiff differential equations.