Precision medicine entails the design of therapies that are matched for each individual patient. Thus, predictive modeling of drug responses for specific patients constitutes a significant challenge for personalized therapy. In this article, we consider a review of approaches that have been proposed to tackle the drug sensitivity prediction problem especially with respect to personalized cancer therapy. We first discuss modeling approaches that are based on genomic characterizations alone and further the discussion by including modeling techniques that integrate both genomic and functional information. A comparative analysis of the prediction performance of four representative algorithms, elastic net, random forest, kernelized Bayesian multi-task learning and deep learning, reflecting the broad classes of regularized linear, ensemble, kernelized and neural network-based models, respectively, has been included in the paper. The review also considers the challenges that need to be addressed for successful implementation of the algorithms in clinical practice.