Superlenses, hyperlenses, and metalenses can achieve a resolution beyond that permitted by diffraction, advancing the development of elastic wave imaging. However, a comprehensive review of these lenses is still lacking. In this article, we aim to survey existing lenses with super-resolution capability. A detailed review of various artificially engineered materials used to build these lenses is provided, including negative-index materials, anisotropic materials, and holey-structured materials. The underlying wave physics associated with these functional materials and lenses are elucidated, particularly the capability to manipulate propagating and evanescent waves, which is interpreted through band diagrams and effective parameters. Recent progress on these lenses and the resolution realized in experiments are also presented. In addition, we discuss the characteristics of different lenses, and an insight into gaps between current designs and practical demands is proposed, followed by promising applications. Finally, potential directions and challenges in this field are offered.