Developing optimal drug combinations is one of the central challenges of cancer treatment research: drug combinations are used to treat most types of cancer, and are almost exclusively responsible for cures of advanced cancers. However, historically successful combination therapies were developed empirically, and the mechanistic basis for their efficacy has been largely speculative. I will present experiments, models, and computational analyses of clinical trial data, to investigate the mechanistic basis of clinically successful combination therapies across 12 types of cancer and 30 different therapies. These studies consistently identify the control of cancer heterogeneity between-patients (inter-tumor) and within-patients (intra-tumor) by independently active drugs as critical contributors to the efficacy of combination therapies in human patients. The key approaches for data analysis and modeling in these studies consist of adapting classical pharmacological concepts to the complex situation of populations of cancers with heterogeneous drug sensitivity. We find that supra-additive drug interactions are uncommon in humans among approved combination therapies, and multiple curative regimens are consistent with drug additivity in both experimental measurements and in clinical outcomes. Mathematical descriptions of heterogeneity in cellular or patient populations, and quantitative experimental measurements of how drug combinations address heterogeneity, lead to accurate predictions of clinical trial results for a diverse range of combination therapies, including those with immune checkpoint inhibitors (correlation between observed and expected Progression Free Survival in 14 trials, Pearson r = 0.98, P < 10^-8) and curative chemotherapy regimens for hematological cancers (correlation between observed and expected response rates and cure rates in childhood ALL, Pearson r = 0.99, P < 10^-10). These results have broad significance for the treatment of cancers, for the interpretation of clinical trials, and point to new opportunities to use combination therapies with greater precision.