In recent years, co-clustering has emerged as a powerful data mining tool that can analyze dyadic data connecting two entities. However, almost all existing co-clustering techniques are partitional, and allow individual rows and columns of a data matrix to belong to only one cluster. Several current applications, such as recommendation systems and market basket analysis, can substantially benefit from a mixed membership of rows and columns. In this paper, we present Bayesian co-clustering (BCC) models, that allow a mixed membership in row and column clusters. BCC maintains separate Dirichlet priors for rows and columns over the mixed membership and assumes each observation to be generated by an exponential family distribution corresponding to its row and column clusters. We propose a fast variational algorithm for inference and parameter estimation. The model is designed to naturally handle sparse matrices as the inference is done only based on the nonmissing entries. In addition to finding a co-cluster structure in observations, the model outputs a low dimensional coembedding, and accurately predicts missing values in the original matrix. We demonstrate the efficacy of the model through experiments on both simulated and real data.