Several probability distributions for circular data have been discussed by Rao (Linear statistical inference and its applications, 2nd edn. Wiley, New York, 1973) in his classic book. The aim of this paper is to introduce model-based clustering methods for bivariate circular or toroidal data. Here a mixture model approach based on the joint distribution of the two dependent circular variables is proposed. In particular, two types of such mixture models are constructed, one based on the marginal and the other on the conditional specification. Convergence property of Expectation-Maximization (EM) algorithm for the members of the regular exponential family used for our models is studied. Cluster properties, such as optimum number, homogeneity, etc. are also discussed. A real life application on gene data is made to illustrate the use of the proposed approaches. Comparison of the two models is also done based on this example. Clustering method for observations on the torus do not seem to be available in the literature, and this paper is possibly the maiden attempt in that direction.