This paper explores second-order viscous hydrodynamics in conformal field theories at finite temperature, focusing on the constraints imposed by conformal invariance. The authors derive the consequences of conformal symmetry for hydrodynamics, classify all second-order terms consistent with conformal symmetry, and compute three out of five new transport coefficients in the strongly coupled $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory using the AdS/CFT correspondence. They also discuss how these new coefficients can arise within the kinetic theory of weakly coupled conformal plasmas. The paper highlights that the Müller-Israel-Stewart theory, commonly used in numerical simulations, does not contain all allowed second-order terms and often lacks terms required by conformal invariance. The authors conclude by analyzing the implications of their findings for the Müller-Israel-Stewart theory and summarize their main results.This paper explores second-order viscous hydrodynamics in conformal field theories at finite temperature, focusing on the constraints imposed by conformal invariance. The authors derive the consequences of conformal symmetry for hydrodynamics, classify all second-order terms consistent with conformal symmetry, and compute three out of five new transport coefficients in the strongly coupled $\mathcal{N}=4$ supersymmetric Yang-Mills (SYM) theory using the AdS/CFT correspondence. They also discuss how these new coefficients can arise within the kinetic theory of weakly coupled conformal plasmas. The paper highlights that the Müller-Israel-Stewart theory, commonly used in numerical simulations, does not contain all allowed second-order terms and often lacks terms required by conformal invariance. The authors conclude by analyzing the implications of their findings for the Müller-Israel-Stewart theory and summarize their main results.