Cancer is a genetic disease induced by mutations in DNA, in particular point mutations in important driver genes that lead to protein malfunctioning and ultimately to tumorigenesis. Screening for the most common DNA point mutations, especially in such genes as TP53, BRCA1 and BRCA2, EGFR, KRAS, or BRAF, is crucial to determine predisposition risk for cancer or to predict response to therapy. In this review, we briefly depict how these genes are involved in cancer, followed by a description of the most common techniques routinely applied for their analysis, including high-throughput next-generation sequencing technology and less expensive low-throughput options, such as real-time PCR, restriction fragment length polymorphism, or high resolution melting analysis. We then introduce benefits of electrochemical biosensors as interesting alternatives to the standard methods in terms of cost, speed, and simplicity. We describe most common strategies involved in electrochemical biosensing of point mutations, relying mostly on PCR or isothermal amplification techniques, and critically discuss major challenges and obstacles that, until now, prevented their more widespread application in clinical settings.