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Collaborative filtering (CF), particularly matrix factorization (MF) based methods, have been widely used in recommender systems. The literature has reported that matrix factorization methods often produce superior accuracy of rating prediction in recommender systems. However, existing matrix factorization methods rarely consider confidence of the rating prediction and thus cannot support advanced recommendation tasks. In this paper, we propose a Confidence-aware Matrix Factorization (CMF) framework to simultaneously optimize the accuracy of rating prediction and measure the prediction confidence in the model. Specifically, we introduce variance parameters for both users and items in the matrix factorization process. Then, prediction interval can be computed to measure confidence for each predicted rating. These confidence quantities can be used to enhance the quality of recommendation results based on Confidence-aware Ranking (CR). We also develop two effective implementations of our framework to compute the confidence-aware matrix factorization for large-scale data. Finally, extensive experiments on three real-world datasets demonstrate the effectiveness of our framework from multiple perspectives.