Mutational signature analysis is a recent computational approach for interpreting somatic mutations in the genome. Its application to cancer data has enhanced our understanding of mutational forces driving tumorigenesis and demonstrated its potential to inform prognosis and treatment decisions. However, methodological challenges remain for discovering new signatures and assigning proper weights to existing signatures, thereby hindering broader clinical applications. Here we present Mutational Signature Calculator (MuSiCal), a rigorous analytical framework with algorithms that solve major problems in the standard workflow. Our simulation studies demonstrate that MuSiCal outperforms state-of-the-art algorithms for both signature discovery and assignment. By reanalyzing more than 2,700 cancer genomes, we provide an improved catalog of signatures and their assignments, discover nine indel signatures absent in the current catalog, resolve long-standing issues with the ambiguous 'flat' signatures and give insights into signatures with unknown etiologies. We expect MuSiCal and the improved catalog to be a step towards establishing best practices for mutational signature analysis.