Preserving native and engineered tissues for long periods without losing biological function is an important and challenging problem in biomedicine. Preservation is necessary to ensure an adequate supply of both native and engineered tissues for transplantation or other use. However, currently available preservation techniques have storage limitations which leave a gap between supply and demand for these tissues. Improved preservation techniques would allow more effective storage and banking as well as safe transportation of native/engineered tissues, and provide off-the-shelf availability of tissue-engineered products to surgeons for implantation. Several preservation methods have been developed and are used currently-hypothermic preservation, cryopreservation, and vitrification. In addition freeze-drying may play a role for some tissue preservation approaches in the future although many challenges remain. Hypothermic preservation is currently employed as the primary clinical organ preservation technique, but the storage time limit is usually measured in hours. Cryopreservation and vitrification use cooling (with or without phase change) to very low temperatures in the presence of high concentrations of cryoprotective chemical agents (CPAs) for preserving tissues. Even though these cryogenic techniques provide longer preservation time than hypothermic preservation, they can induce freezing injuries in tissue due to ice-formation within cellular or extracellular spaces, and/or by the addition and removal of high concentrations of CPAs. In this review, challenges in tissue preservation and corresponding engineering efforts to address them are reviewed. These include (1) biophysics and mass transfer during CPA loading/unloading, and freezing/thawing; (2) heat transfer including phase change behavior and thermal properties of tissues in the presence of CPAs, and heat transfer models during preservation; and (3) changes in mechanical properties of tissues after preservation.