The aim of this review will be to provide a basic understanding of the biophysical processes that accompany the application of cryopreservation in reproductive medicine. The ability to store cells in 'suspended animation' outside the body has become a keystone practice in the development of many modern clinical therapies, and, in fact, the sciences of cryobiology and IVF have developed in parallel over the past 50 years. During this time, some of the underlying principles of the quantitative biophysical aspects of cryobiology have been clarified. Water is the universal biocompatible solvent, but also possesses unique properties for stability of living cells. Whilst low temperatures themselves have defined effects on cell structure and function, it is the phase transition of water to ice that is the most profound challenge for survival. The thermodynamics of dilute aqueous solutions dictate how cells and tissues respond to the freezing process. Current concepts of nucleation, ice crystal growth and solute exclusion from the ice lattice will be discussed to illustrate what cells must negotiate to avoid lethal damage, and the role of cryoprotectants in enhancing recovery. Quantitative formalisms now exist to model and predict how water and solutes move across cell membranes before and during freezing, or how nucleation events will proceed, and these will be outlined. Cryoprotectants have both positive and negative effects on cell function depending on the kinetics of exposure. The concept of tolerable osmotic excursion of cell volume will be discussed, along with the evidence for a 'pseudo-glassy' state for cells during traditional cryopreservation. This will be compared with the recent interest in promoting glassy states in the whole sample using vitrification protocols, outlining the advantages and drawbacks of each approach. Additional methods for controlling ice nucleation have a role to play here, and a brief outline of current technologies will be given. Finally, issues of safety and stability of cryopreserved samples will be set out.