In this paper, the identifiability of discrete-time Affine Linear Parameter-Varying (ALPV) models is studied. Examples are presented to show that, in general, the identifiability of ALPV model parameterizations does not guarantee the identifiability of the LTI parameterizations composed of frozen LTI models. A new sufficient and necessary condition is then introduced in order to guarantee the structural identifiability for ALPV parameterizations. The identifiability of this class of parameterizations is related to the lack of state space isomorphisms between any two models corresponding to different parameter values. In addition, we present a sufficient and necessary condition for local structural identifiability, and a sufficient condition for (global) structural identifiability which are both based on the rank of a user-defined matrix. These latter conditions allow systematic verification of identifiability. Numerical examples are finally presented to illustrate our results. (C) 2017 Elsevier Ltd. All rights reserved.