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We extend the Data-Driven formulation of problems in elasticity of Kirchdoerfer and Ortiz [1] to inelasticity. This extension differs fundamentally from Data-Driven problems in elasticity in that the material data set evolves in time as a consequence of the history dependence of the material. We investigate three representational paradigms for the evolving material data sets: i) materials with memory, i. e., conditioning the material data set to the past history of deformation; ii) differential materials, i. e., conditioning the material data set to short histories of stress and strain; and iii) history variables, i. e., conditioning the material data set to ad hoc variables encoding partial information about the history of stress and strain. We also consider combinations of the three paradigms thereof and investigate their ability to represent the evolving data sets of different classes of inelastic materials, including viscoelasticity, viscoplasticity and plasticity. We present selected numerical examples that demonstrate the range and scope of Data-Driven inelasticity and the numerical performance of implementations thereof.