Semitoric systems are a type of 4-dimensional integrable system which has been classified by Pelayo-Vu Ngoc in terms of five invariants, one of which is a family of polygons generalizing the Delzant polygons which classify 4-dimensional toric integrable systems. In this talk we present one-parameter families of integrable systems which are semitoric at all but finitely many values of the parameter, which we call semitoric families, with the goal of developing a strategy to find a semitoric system associated to a given partial list of semitoric invariants. Using these families, we find several new explicit semitoric systems which display various behavior and are of importance in the semitoric minimal models program. This work is joint with Y. Le Floch (and work joint with S. Hohloch, D. Kane, and A. Pelayo will also be mentioned).