This chapter focuses on a number of relevant models of time-dependent and quasistatic process of contact. It describes the material's behavior with an elastic, viscoelastic or viscoplastic constitutive law. The contact is with normal compliance with or without unilateral constraint and, for most of the models, it is frictionless. The chapter also presents the classical formulation for each model and provides a short description of the equations and boundary conditions. It lists the assumptions on the data and derives a variational formulation which is in a form of a history-dependent variational-hemivariational inequality for the displacement field. A mathematical model of a time-dependent process of contact combines the constitutive law, the displacement boundary condition, the traction boundary condition and specific contact conditions. A careful analysis of the contact model reveals that the main ingredient in its study consists in arguments of history dependent operators.