srjecsSRJECSSRJECS2788-9408https://doi.org/10.47310/srjecs.2026.v06i01.007Recent Advances in β-Tilting Theory and its Generalizations in Module CategoriesThe mathematical -tilting theory idea (the mathematical -t theory) originated from the conceptual paintings of Adachi and his colleagues in 2014 [1], and it rapidly emerged as a primary focus of investigation within representation theory(R-theory) of finite-dimensional algebras (F-D algebras ). Integrating the idea of ​​tilt, this framework presents an efficient combinatorial and isomorphic tool for reading rotation training, silt complexes, and cluster- tilting (CT) objects within modular classes .This study provides a systematic and self-contained introduction to -t theory , especially designed for readers with a standard background in representation theory. It aims to enable researchers to grasp the fundamental concepts without the need for extensive reference to external sources, while maintaining full commitment to high mathematical rigor. In addition, the study unifies silting theory and cluster-tilting theory (CT theory) within a common perspective, gathering in one place the basic definitions, important bijections, and mutation techniques that form the core of this field [1]. In addition to the classical foundations, the research criticizes major developments published between 2023 and 2026, including new properties of -tilting ( -t) finiteness for Borel–Schur algebras and group algebras of generalized symmetric groups, and their applications to Frobenius–Perron dimensions and generalized preprojective algebras [2–8]. Particular attention is given to recent developments in higher torsion classes, d-tilting theory ( d-t-theory) and duplicated algebras. The study concludes by highlighting several key open problems such as the classification of minimal -tilting infinite algebras ( -t-i algebras ) and the explicit description of the vital bijections for important families of algebras which continue to motivate current research [9].This work aims to be an accessible entry point for beginners and a comprehensive reference for active researchers in representation theory ( R- theory ) and related fields [1,2].