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In recent years, many researchers have defined new sequence spaces whose Cesàro, Riesz, B(r,s), etc., averages are almost convergent and have introduced some of the algebraic and topological properties of these new sequence spaces. Besides, in addition to matrix transformation problems, researchers deal with core problems. After, the definition of ordinary, it means classical convergence of sequences, G. G. Lorentz offered the definition of almost convergence used in this work frequently by means of Banach limits and proposed that the sequence space of almost convergent sequences is not a matrix domain of any matrix. In this book, out of the meaning of the ordinary almost convergence, we deal with a convergence idea that is more general and comprehensive than the ordinary almost convergence. Firstly, to advance the idea of ordinary almost convergence, we describe the set of all T- convergent sequences, after by considering this set we give a new convergence definition by taking Riesz matrix instead of matrix T and we build the some of the new sequence spaces and investigate some important properties on these concepts.