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In present work, we describe and investigate torsion theoretic versions of δ‐supplemented modules via a hereditary torsion theory τ. With this aim, first, we define δ_τ‐small submodules. On this basis, the concepts of δ_τ‐lifting modules, δ_τ‐supplemented modules, and amply δ_τ‐supplemented modules and their fundamental properties are given, respectively. Furthermore, we present δ_τ‐semiperfect modules and give a characterization for them via (amply) δ_τ‐supplemented modules. Even we supply binary relations between these new module classes.