Summary: The identifiability problem arises when a parametric family of probability distributions is not related to the space of parameters by a one-to-one correspondence. We give a brief review of the recent theoretical developments relating to the identifiability problems, specifically focussing on the equivalence classes, conditions for identifiability and determination of identifying functions which lead to reparametrization. We present general definitions of identifiable constraints and model-preserving constraints, and indicate how any given parametric constraint can be expressed as an intersection of an identifiable constraint and a model-preserving constraint. We also present a necessary and sufficient condition for a linear constraint in a linear model to be model-preserving.