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We formulate a Kalman-style realization theory for discrete-time affine LPV systems. We characterize those input-output behaviors which exactly correspond to affine LPV systems. In addition, we present necessary and sufficient conditions for minimality of affine LPV systems and show that equivalent minimal realizations are unique up to isomorphism. The results are derived by reducing the problem to the realization problem for linear switched systems. In addition, we show that an input-output map has a realization by an affine LPV system if and only if it satisfies certain types of input-output equations.