The paper by N. Reshetikhin and V.G. Turaev aims to construct new topological invariants for compact oriented 3-manifolds and framed links within these manifolds. The invariant for a closed oriented 3-manifold is a sequence of complex numbers parametrized by roots of unity, with the terms for a framed link in \( S^3 \) being the values of the Jones polynomial evaluated at these roots. For manifolds with boundaries, the invariant is a finite-dimensional complex linear operator, leading to a 3-dimensional topological quantum field theory (TQFT) associated with each root of unity \( q \). This TQFT associates a finite-dimensional complex linear space with every closed oriented surface and defines a projective action of the modular group on this space. The constructions are inspired by E. Witten's work on quantum field theory and the nonabelian Chern-Simons action, which defined invariants of 3-manifolds and links. The authors use a practical approach, reducing the general case to links in \( S^3 \) and applying the classical Jones polynomial and related invariants. They also introduce the concept of modular Hopf algebras, showing that each such algebra gives rise to a 3-dimensional TQFT, producing numerical invariants of closed oriented 3-manifolds and links. The paper is organized into several sections, covering ribbon Hopf algebras, modular Hopf algebras, operator invariants of 3-dimensional cobordisms, and proofs of theorems. The authors acknowledge various contributions from other researchers and hospitality from institutions where parts of the work were conducted.The paper by N. Reshetikhin and V.G. Turaev aims to construct new topological invariants for compact oriented 3-manifolds and framed links within these manifolds. The invariant for a closed oriented 3-manifold is a sequence of complex numbers parametrized by roots of unity, with the terms for a framed link in \( S^3 \) being the values of the Jones polynomial evaluated at these roots. For manifolds with boundaries, the invariant is a finite-dimensional complex linear operator, leading to a 3-dimensional topological quantum field theory (TQFT) associated with each root of unity \( q \). This TQFT associates a finite-dimensional complex linear space with every closed oriented surface and defines a projective action of the modular group on this space. The constructions are inspired by E. Witten's work on quantum field theory and the nonabelian Chern-Simons action, which defined invariants of 3-manifolds and links. The authors use a practical approach, reducing the general case to links in \( S^3 \) and applying the classical Jones polynomial and related invariants. They also introduce the concept of modular Hopf algebras, showing that each such algebra gives rise to a 3-dimensional TQFT, producing numerical invariants of closed oriented 3-manifolds and links. The paper is organized into several sections, covering ribbon Hopf algebras, modular Hopf algebras, operator invariants of 3-dimensional cobordisms, and proofs of theorems. The authors acknowledge various contributions from other researchers and hospitality from institutions where parts of the work were conducted.