We propose a Bayesian response-adaptive covariate-balanced (RC) randomization design for multiple-arm comparative clinical trials. The goal of the design is to skew the allocation probability to more efficacious treatment arms, while also balancing the distribution of the covariates across the arms. In particular, we first propose a new covariate-adaptive randomization (CA) method based on a prognostic score that naturally accommodates continuous and categorical prognostic factors and automatically assigns imbalance weights to covariates according to their importance in response prediction. We then incorporate this CA design into a group sequential response-adaptive randomization (RA) scheme. The resulting RC randomization design combines the advantages of both CA and RA randomizations and meets the design goal. We illustrate the proposed design through its application to a phase II leukemia clinical trial, and evaluate its operating characteristics through simulation studies. Copyright © 2011 John Wiley & Sons, Ltd.