Papillary renal cell carcinoma (pRCC) is an important subtype of kidney cancer with a problematic pathological classification and highly variable clinical behaviour. Here we sequence the genomes or exomes of 31 pRCCs, and in four tumours, multi-region sequencing is undertaken. We identify BAP1, SETD2, ARID2 and Nrf2 pathway genes (KEAP1, NHE2L2 and CUL3) as probable drivers, together with at least eight other possible drivers. However, only ~10% of tumours harbour detectable pathogenic changes in any one driver gene, and where present, the mutations are often predicted to be present within cancer sub-clones. We specifically detect parallel evolution of multiple SETD2 mutations within different sub-regions of the same tumour. By contrast, large copy number gains of chromosomes 7, 12, 16 and 17 are usually early, monoclonal changes in pRCC evolution. The predominance of large copy number variants as the major drivers for pRCC highlights an unusual mode of tumorigenesis that may challenge precision medicine approaches.