We analytically study the damping of Alfven mode oscillations in the chromosphere and in coronal loops. In the partially ionized chromosphere the dominant damping process of Alfven waves is due to collisions between ions and neutrals. We calculate the damping time for Alfven waves of a given frequency, propagating through model chromospheres of various solar structures such as active region plage, quiet sun, and the penumbra and umbra of sunspots. For a given wave frequency, the maximum damping always occurs at temperature minimum heights and in the coldest structure(s), i.e., the umbra of sunspots. Energy dissipation due to ion-neutral damping of Alfven waves with an energy flux of 107 ergs cm-3 s- 1 can play a considerable role in the energy balance of umbrae, quiet sun, and plage for Alfven wave periods of the order, respectively, 50, 5, and 0.5 s. We also consider Alfven waves in coronal loops and the leakage of wave energy through the footpoints. We assume a three-layer model of coronal loops with constant Alfven speed vA (and no damping) in the corona, vA varying exponentially with height in the dissipative chromosphere, and vA again constant in the photosphere at the end of the loop. We find an exact analytical solution in the chromospheric part. Using these solutions, we estimate the leakage of wave energy from the coronal volume through the footpoint regions of the loop and find that the presence of a moderate amount of chromospheric damping can enhance the footpoint leakage. We apply this result to determine the damping time of standing waves in coronal loops. The enhanced footpoint leakage also has implications for theories of coronal heating based on resonant absorption. Finally, we find exact expressions for the damping of Alfven waves launched in the photosphere and upward propagating through the chromosphere and into the corona. The partially ionized chromosphere presents an effective barrier for upward propagating Alfven waves with periods less than a few seconds.