Mathematical physicists have studied degenerations of Lie groups and their representations, which they call contractions. In my talk I will present a new approach for studying these contractions within the framework of algebraic families of Harish-Chandra modules. A construction of natural families that incorporates both a real reductive group and its compact form will be given. As a running example, I will consider the family associated with SL2(R) and describe the classification of its irreducible families of Harish-Chandra modules. If time permits, I will construct the Mackey bijection between the admissible duals of SL2(R) and its Cartan motion group via algebraic families of Harish-Chandra modules. This is a joint work with Joseph Bernstein and Nigel Higson.