Identifying the genes and mutations that drive the emergence of tumors is a critical step to improving our understanding of cancer and identifying new directions for disease diagnosis and treatment. Despite the large volume of genomics data, the precise detection of driver mutations and their carrying genes, known as cancer driver genes, from the millions of possible somatic mutations remains a challenge. Computational methods play an increasingly important role in discovering genomic patterns associated with cancer drivers and developing predictive models to identify these elements. Machine learning (ML), including deep learning, has been the engine behind many of these efforts and provides excellent opportunities for tackling remaining gaps in the field. Thus, this survey aims to perform a comprehensive analysis of ML-based computational approaches to identify cancer driver mutations and genes, providing an integrated, panoramic view of the broad data and algorithmic landscape within this scientific problem. We discuss how the interactions among data types and ML algorithms have been explored in previous solutions and outline current analytical limitations that deserve further attention from the scientific community. We hope that by helping readers become more familiar with significant developments in the field brought by ML, we may inspire new researchers to address open problems and advance our knowledge towards cancer driver discovery.