Identiiability is a necessary condition for the existence of consistent estimates for the parameters of mixture models. In this paper the identiiability of nite mixtures of linear regression models with Normal errors is investigated. Three diierent models are treated: Mixture models with random and xed independent variables and a model with xed partition of the data to the mixture components. Sometimes only parts of the unknown parameter values are of interest. \Partial identiiability" is introduced for this purpose. It turns out that identiiability of nite linear regression mixtures depends on the number of p ? 1-dimensional hyperplanes which one needs to cover the independent variables. Counterexamples and suu-cient conditions for identiiability are given for all models.