The bivariate Sinh-Elliptical (BSE) distribution is a generalization of the well-known Rieck's (1989) Sinh-Normal distribution that is quite useful in Birnbaum-Saunders (BS) regression model. The main aim of this paper is to define the BSE distribution and discuss some of its properties, such as marginal and conditional distributions and moments. In addition, the asymptotic properties of method of moments estimators are studied, extending some existing theoretical results in the literature. These results are obtained by using some known properties of the bivariate elliptical distribution. This development can be viewed as a follow-up to the recent work on bivariate Birnbaum-Saunders distribution by Kundu et al. (2010) towards some applications in the regression setup. The measurement error models are also introduced as part of the application of the results developed here. 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