Primary hyperparathyroidism (PHPT) is a common endocrine disorder characterized by dysregulation of parathyroid hormone release. The large majority of PHPT cases are attributable to sporadic, single-gland parathyroid adenoma, in which MEN1 and CCND1/cyclin D1 are the most well-established drivers of tumorigenesis. Sporadic parathyroid carcinoma, which appears to mostly arise through molecular pathways distinct from those causing benign parathyroid tumors, is rare and is most frequently driven by mutational inactivation of the CDC73 (HRPT2) tumor suppressor gene. Targeted investigation of suspected tumor driver genes, as well as unbiased whole-genome or exome sequencing of small cohorts, have revealed additional novel candidate tumor genes in sporadic parathyroid neoplasia, generally at modest or low mutational frequencies consistent with marked molecular genetic heterogeneity from tumor to tumor. The ability of these additional candidates to participate in the pathogenic process of driving parathyroid tumorigenesis in vivo largely remains to be demonstrated experimentally. This review will summarize the molecular genetic abnormalities identified to date in sporadic PHPT and discuss the strength of evidence for their proposed roles in parathyroid tumor formation.