In this chapter we introduce the notion of consistency for interacting particle systems. A system is consistent if the action of removing at random a particle commutes with the evolution. Beyond considering the problem at the level of particle configurations we will analyze the notion of consistency from the point of view of (labeled) particle positions. We will show that consistent particle systems satisfy a set of recursion relations for joint factorial moments of particle occupancies. We discuss the relation between consistency and self-duality. We show that adding absorbing sites conserves consistency, which is important in the study of boundary driven systems where the dual is an absorbing system.