The paper explores the quantization of classical integrable systems through four-dimensional N=2 supersymmetric gauge theories in the Ω-background. It establishes a correspondence between the supersymmetric vacua of these gauge theories and the quantum states of integrable systems. The Ω-background introduces a deformation parameter ε, which is identified with the Planck constant. The twisted chiral ring maps to quantum Hamiltonians, and supersymmetric vacua correspond to Bethe states of integrable systems. The low-energy effective theory of the gauge theory is shown to have a two-dimensional twisted superpotential, which becomes the Yang-Yang function of the integrable system. Thermodynamic Bethe ansatz-like formulas are derived for these functions and the spectra of commuting Hamiltonians. The construction is illustrated for various systems, including the periodic Toda chain, elliptic Calogero-Moser system, and their relativistic counterparts. The paper also discusses the quantization of the Hitchin system. The key idea is that the gauge theory provides a framework for quantizing classical integrable systems, with the Ω-background playing a crucial role in this process. The results demonstrate a deep connection between gauge theories and quantum integrable systems, offering new insights into their structure and properties.The paper explores the quantization of classical integrable systems through four-dimensional N=2 supersymmetric gauge theories in the Ω-background. It establishes a correspondence between the supersymmetric vacua of these gauge theories and the quantum states of integrable systems. The Ω-background introduces a deformation parameter ε, which is identified with the Planck constant. The twisted chiral ring maps to quantum Hamiltonians, and supersymmetric vacua correspond to Bethe states of integrable systems. The low-energy effective theory of the gauge theory is shown to have a two-dimensional twisted superpotential, which becomes the Yang-Yang function of the integrable system. Thermodynamic Bethe ansatz-like formulas are derived for these functions and the spectra of commuting Hamiltonians. The construction is illustrated for various systems, including the periodic Toda chain, elliptic Calogero-Moser system, and their relativistic counterparts. The paper also discusses the quantization of the Hitchin system. The key idea is that the gauge theory provides a framework for quantizing classical integrable systems, with the Ω-background playing a crucial role in this process. The results demonstrate a deep connection between gauge theories and quantum integrable systems, offering new insights into their structure and properties.