Recent studies have identified prevalent subclonal architectures within many cancer types. However, the temporal evolutionary dynamics that produce these subclonal architectures remain unknown. Here we measure evolutionary dynamics in primary human cancers using computational modelling of clonal selection applied to high throughput sequencing data. Our approach simultaneously determines the subclonal architecture of a tumour sample, and measures the mutation rate, the selective advantage, and the time of appearance of subclones. Simulations demonstrate the accuracy of the method, and revealed the degree to which evolutionary dynamics are recorded in the genome. Application of our method to high-depth sequencing data from gastric and lung cancers revealed that detectable subclones consistently emerged early during tumour growth and had considerably large fitness advantages (>20% growth advantage). Our quantitative platform provides new insight into the evolutionary history of cancers by facilitating the measurement of fundamental evolutionary parameters in individual patients.