Classic cryonics scene from Futurama
Cryogenics is used in rocket fuels, MRIs, electric power transmission, freezing foods, infrared cameras, blood banks, and of course preserving bodies. As conductive materials get colder, they become more conductive, at a certain point, these materials become "superconductive". Superconductivity is very important for transferring power and creating very powerful magnets, we use it all the time, whether we know it or not. Cryogenics is integral to creating superconductive materials because it is necessary to keep them cold. Cryogenics also has applications in something called "cryomilling". A lot of materials are extremely difficult to mill under standard conditions, but they become brittle at lower temperatures, meaning that cryogenics can make them easier to mill.
Superconductive magnets are much stronger than regular magnets
Cryogenics is divided up into several different fields, cryobiology (the study of the effects of low temperatures on organisms), cryonics (the medical technology to preserve bodies to be revived later), cryosurgery (using low temperature things to kill things like cancer) and cryoelectronics (the study of superconductivity).
The field is growing rapidly, currently much of the research is into the preservation of life, expanding into the realm of aiding heart attack victims among other things.