This paper evaluates the importance of thermoelastic damping as a fundamental dissipation mechanism in micro- and nano-mechanical systems (MEMS and NEMS). The authors derive a simple expression for thermoelastic damping in thin beams and show that Zener's approximation using a Lorentzian with a single thermal relaxation time slightly deviates from the exact expression. Thermoelastic damping is shown to be significant at small scales due to two factors: the damping strength is independent of geometry and the peak damping frequency depends on beam dimensions. The paper discusses the process of thermoelastic damping, its implications for MEMS and NEMS, and presents exact solutions for thin beams. It also compares the exact expression with Zener's approximation and shows that the exact expression is more accurate. The results are applied to GaAs and Silicon, typical materials used in MEMS and NEMS, and the dependence of thermoelastic damping on beam geometry is analyzed. The paper concludes that thermoelastic damping is a significant source of dissipation for MEMS and NEMS at temperatures above 100K, and that the relaxation time depends on the frequency. The paper also discusses the damping of longitudinal waves and the Akhiezer effect. The authors acknowledge the support of DARPA and thank colleagues for their contributions.This paper evaluates the importance of thermoelastic damping as a fundamental dissipation mechanism in micro- and nano-mechanical systems (MEMS and NEMS). The authors derive a simple expression for thermoelastic damping in thin beams and show that Zener's approximation using a Lorentzian with a single thermal relaxation time slightly deviates from the exact expression. Thermoelastic damping is shown to be significant at small scales due to two factors: the damping strength is independent of geometry and the peak damping frequency depends on beam dimensions. The paper discusses the process of thermoelastic damping, its implications for MEMS and NEMS, and presents exact solutions for thin beams. It also compares the exact expression with Zener's approximation and shows that the exact expression is more accurate. The results are applied to GaAs and Silicon, typical materials used in MEMS and NEMS, and the dependence of thermoelastic damping on beam geometry is analyzed. The paper concludes that thermoelastic damping is a significant source of dissipation for MEMS and NEMS at temperatures above 100K, and that the relaxation time depends on the frequency. The paper also discusses the damping of longitudinal waves and the Akhiezer effect. The authors acknowledge the support of DARPA and thank colleagues for their contributions.