Neoplastic diseases of the upper respiratory airways, as well as head and neck cancers, are a frequent cause of death and significantly affect the quality of life of both patients and survivors. As the frequency increases, new and improved treatment techniques are sought. Promising properties in this respect are expressed by a natural compound – curcumin. Along with its derivatives, it was found useful in the treatment of a series of cancers. Curcumin was found to be effective in clinical trials and in vitro, in vivo anticancer experiments. Nanoformulations (e.g., poly(lactide-co-glycolic acid)-based nanoparticles, nanoemulsions), and modifications of curcumin, as well as its combinations with other substances (e.g., catechins, cisplatin) or treatments (e.g., radiotherapy or local use in inhalation), were found to enhance the antitumor effect. This review aims to summarize the recent findings for the treatment of head and neck diseases, especially squamous cell carcinomas (HNSCCs), including drawing attention to the constant use of the misidentified Hep-2 cell line and proposing databases purposed at eliminating this problem. Moreover, this manuscript focuses on pointing out the molecular mechanisms of therapy that have been reached and emphasizing the shortcomings that still need to be addressed.