This Volume Looks At The Study Of Dynamical Systems With Discontinuities. Discontinuities Arise When Systems Are Subject To Switches, Decisions, Or Other Abrupt Changes In Their Underlying Properties That Require A 'non-smooth' Definition. A Review Of Current Ideas And Introduction To Key Methods Is Given, With A View To Opening Discussion Of A Major Open Problem In Our Fundamental Understanding Of What Nonsmooth Models Are.what Does A Nonsmooth Model Represent: An Approximation, A Toy Model, A Sophisticated Qualitative Capturing Of Empirical Law, Or A Mere Abstraction?tackling This Question Means Confronting Rarely Discussed Indeterminacies And Ambiguities In How We Define, Simulate, And Solve Nonsmooth Models.