Meta-structures, including metamaterials and metasurfaces, possess remarkable physical properties beyond those observed in natural materials and thus have exhibited unique wave manipulation abilities ranging from quantum to classical transports. The past decades have witnessed the explosive development and numerous implications of meta-structures in elastic-wave control under the Hermitian condition. However, more notably, a lot of recent research has been made to show that non-Hermitian meta-structures offer novel means for wave manipulation. Non-Hermiticity has enhanced both the accuracy and efficiency of wave steering capabilities. To this end, starting from electromagnetics and acoustics, we mainly review the up-to-date progress of non-Hermitian elastic meta-structures with a focus on their extraordinary elastic-wave control. A variety of promising scenarios realized by non-Hermitian elastic metamaterials and metasurfaces, such as the parity-time-symmetric system and the skin effect, are summarized. Furthermore, the perspectives and challenges of non-Hermitian elastic meta-structures for future key opportunities are outlined.