With data from recent large-scale drug sensitivity measurement campaigns, it is now possible to build and test models predicting responses for more than one hundred anticancer drugs against several hundreds of human cancer cell lines. Traditional quantitative structure-activity relationship (QSAR) approaches focus on small molecules in searching for their structural properties predictive of the biological activity in a single cell line or a single tissue type. We extend this line of research in two directions: (1) an integrative QSAR approach predicting the responses to new drugs for a panel of multiple known cancer cell lines simultaneously and (2) a personalized QSAR approach predicting the responses to new drugs for new cancer cell lines. To solve the modeling task, we apply a novel kernelized Bayesian matrix factorization method. For maximum applicability and predictive performance, the method optionally utilizes genomic features of cell lines and target information on drugs in addition to chemical drug descriptors. In a case study with 116 anticancer drugs and 650 cell lines, we demonstrate the usefulness of the method in several relevant prediction scenarios, differing in the amount of available information, and analyze the importance of various types of drug features for the response prediction. Furthermore, after predicting the missing values of the data set, a complete global map of drug response is explored to assess treatment potential and treatment range of therapeutically interesting anticancer drugs.