The problem of multimodal clustering arises whenever the data are gathered with several physically different sensors. Observations from different modalities are not necessarily aligned in the sense that there is no obvious way to associate or to compare them in some common space. A solution may consist in considering multiple clustering tasks independently for each modality. The main difficulty with such an approach is to guarantee that the unimodal clusterings are mutually consistent. In this paper we show that multimodal clustering can be addressed within a novel framework, namely conjugate mixture models. These models exploit the explicit transformations that are often available between an unobserved parameter space (objects) and each one of the observation spaces (sensors). We formulate the problem as a likelihood maximization task and we derive the associated conjugate expectation-maximization algorithm. The convergence properties of the proposed algorithm are thouroughly investigated. Several local/global optimization techniques are proposed in order to increase its convergence speed. Two initialization strategies are proposed and compared. A consistent model-selection criterion is proposed. The algorithm and its variants are tested and evaluated within the task of 3D localization of several speakers using both auditory and visual data.