A design of robust error-correcting codes that achieve reliable communication over various channels is termed universal. The universality of spatially coupled generalized low-density parity-check (SC-GLDPC) codes over memoryless symmetric channels is analyzed in this paper. Moreover, an extrinsic information transfer recursion approach is developed to prove the universality of SC-GLDPC codes over the binary erasure channel and the additive white Gaussian noise channel, and the robustness of SC-GLDPC codes to impulsive channels with symmetric alpha-stable noise is demonstrated through the consistency of the required scale parameter to provide a specified bit error rate. Numerical results show that, SC-GLDPC codes can fulfill the universality over these channels.