This paper introduces new topological invariants for compact oriented 3-manifolds and framed links within them. The invariants are sequences of complex numbers parameterized by roots of unity. For framed links in S³, these values correspond to the Jones polynomial evaluated at roots of unity. The invariants for 3-manifolds with boundary are finite-dimensional complex linear operators, leading to 3D topological quantum field theories. These theories associate complex linear spaces with closed oriented surfaces and include projective actions of the modular group. The work is inspired by Witten's quantum field theory based on Chern-Simons action, suggesting a mathematical counterpart. The authors use surgery to reduce general 3-manifolds to S³, applying the Jones polynomial and related link invariants. The Jones polynomial is connected to the quantum enveloping algebra of sl₂(C) and its representations, providing a rich set of link invariants. The paper introduces modular Hopf algebras, which yield topological quantum field theories. For each root of unity q, the quantum deformation of sl₂ gives a finite-dimensional modular Hopf algebra, producing the discussed invariants. The paper is structured into sections covering ribbon Hopf algebras, modular Hopf algebras, operator invariants of cobordisms, and proofs of theorems. It also discusses modular Hopf algebras associated with the quantum deformation of sl₂. The authors acknowledge contributions from various mathematicians and note that parts of the work were conducted during visits to research institutions.This paper introduces new topological invariants for compact oriented 3-manifolds and framed links within them. The invariants are sequences of complex numbers parameterized by roots of unity. For framed links in S³, these values correspond to the Jones polynomial evaluated at roots of unity. The invariants for 3-manifolds with boundary are finite-dimensional complex linear operators, leading to 3D topological quantum field theories. These theories associate complex linear spaces with closed oriented surfaces and include projective actions of the modular group. The work is inspired by Witten's quantum field theory based on Chern-Simons action, suggesting a mathematical counterpart. The authors use surgery to reduce general 3-manifolds to S³, applying the Jones polynomial and related link invariants. The Jones polynomial is connected to the quantum enveloping algebra of sl₂(C) and its representations, providing a rich set of link invariants. The paper introduces modular Hopf algebras, which yield topological quantum field theories. For each root of unity q, the quantum deformation of sl₂ gives a finite-dimensional modular Hopf algebra, producing the discussed invariants. The paper is structured into sections covering ribbon Hopf algebras, modular Hopf algebras, operator invariants of cobordisms, and proofs of theorems. It also discusses modular Hopf algebras associated with the quantum deformation of sl₂. The authors acknowledge contributions from various mathematicians and note that parts of the work were conducted during visits to research institutions.