The N⁶-Methyladenosine (m⁶A) modification of RNA transcripts is the most prevalent and abundant internal modification in eukaryotic messenger RNAs (mRNAs) and plays diverse and important roles in normal biological processes. Extensive studies have indicated that dysregulated m⁶A modification and m⁶A-associated proteins play critical roles in tumorigenesis and cancer progression. However, m⁶A-mediated physiological consequences often lead to opposite outcomes in a biological context-dependent manner. Therefore, context-related complexity must be meaningfully considered to obtain a comprehensive understanding of RNA methylation. Recently, it has been reported that m⁶A-modified RNAs are closely related to the regulation of the DNA damage response and genomic integrity maintenance. Here, we present an overview of the current knowledge on the m⁶A modification and its function in human cancer, particularly in relation to the DNA damage response and genomic instability.