The Cosmic Linear Anisotropy Solving System (CLASS) is a new Boltzmann code that improves speed and precision in cosmological parameter estimation. It incorporates three key approximation schemes: the tight coupling approximation (TCA), the ultra-relativistic fluid approximation (UFA), and the radiation streaming approximation (RSA). The TCA simplifies the baryon-photon coupling equations, allowing for first-, second-, or a compromise between these orders. The UFA approximates ultra-relativistic fluids, such as massless neutrinos, by truncating the Boltzmann hierarchy and using fluid equations. The RSA accounts for reionization in radiation streaming, improving the accuracy of temperature and polarisation spectra. CLASS is designed to be flexible and efficient, with a stiff integrator that enhances performance even in stiff systems. The code's TCA implementation allows for different orders of approximation, with second-order schemes providing better accuracy. The UFA reduces computational complexity by truncating the hierarchy at low multipoles, making it efficient for large-scale structures. The RSA improves the treatment of radiation streaming, particularly during reionization, leading to more accurate predictions of CMB and matter power spectra. The paper compares these approximations with existing codes like CAMB, showing that CLASS achieves higher accuracy and efficiency. The TCA schemes, especially second-order ones, provide better results than first-order approximations. The UFA and RSA further enhance the code's performance by reducing computational load and improving accuracy in specific regimes. CLASS is optimized for cosmological parameter estimation, offering a balance between speed and precision, making it a valuable tool for analyzing CMB and large-scale structure data.The Cosmic Linear Anisotropy Solving System (CLASS) is a new Boltzmann code that improves speed and precision in cosmological parameter estimation. It incorporates three key approximation schemes: the tight coupling approximation (TCA), the ultra-relativistic fluid approximation (UFA), and the radiation streaming approximation (RSA). The TCA simplifies the baryon-photon coupling equations, allowing for first-, second-, or a compromise between these orders. The UFA approximates ultra-relativistic fluids, such as massless neutrinos, by truncating the Boltzmann hierarchy and using fluid equations. The RSA accounts for reionization in radiation streaming, improving the accuracy of temperature and polarisation spectra. CLASS is designed to be flexible and efficient, with a stiff integrator that enhances performance even in stiff systems. The code's TCA implementation allows for different orders of approximation, with second-order schemes providing better accuracy. The UFA reduces computational complexity by truncating the hierarchy at low multipoles, making it efficient for large-scale structures. The RSA improves the treatment of radiation streaming, particularly during reionization, leading to more accurate predictions of CMB and matter power spectra. The paper compares these approximations with existing codes like CAMB, showing that CLASS achieves higher accuracy and efficiency. The TCA schemes, especially second-order ones, provide better results than first-order approximations. The UFA and RSA further enhance the code's performance by reducing computational load and improving accuracy in specific regimes. CLASS is optimized for cosmological parameter estimation, offering a balance between speed and precision, making it a valuable tool for analyzing CMB and large-scale structure data.