This paper has been addressed to the well-known problem of energy in gravitational waves. We have investigated the energy of cylindrical gravitational waves in the context of General Relativity and teleparallel theory of gravity. For this purpose, the prescriptions of Einstein, Landau-Lifshitz, Bergmann-Thomson, and Möller are used in both the theories. It is shown that these energy-momentum complexes do not provide equivalent results in the two theories. However, these turn out to be constant for all the prescriptions except Möller in both the theories at large distances.