This review focuses on the molecular biology of head and neck squamous cell carcinomas and presents current and emerging biomarkers of the response of patients to induction chemotherapy. The usefulness of genes, proteins, and parameters from diagnostic clinical imaging as well as other clinicopathological parameters is thoroughly discussed. The role of induction chemotherapy before radiotherapy or before chemo-radiotherapy is still debated, as the data on its efficacy are somehow confusing. Despite the constant improvement of treatment protocols and the introduction of new cytostatics, there is still no consensus regarding the use of induction chemotherapy in the treatment of head and neck cancer, with the possible exception of larynx preservation. Such difficulties indicate that potential future treatment strategies should be personalized. Personalized medicine, in which individual tumor genetics drive the selection of targeted therapies and treatment plans for each patient, has recently emerged as the next generation of cancer therapy. Early prediction of treatment outcome or its toxicity may be highly beneficial for those who are at risk of the development of severe toxicities or treatment failure—a different treatment strategy may be applied to these patients, sparing them unnecessary pain. The literature search was carried out in the PubMed and ScienceDirect databases as well as in the selected conference proceedings repositories. Of the 265 articles and abstracts found, only 30 met the following inclusion criteria: human studies, analyzing prediction of induction chemotherapy outcome or toxicity based on the pretreatment (or after the first cycle, if more cycles of induction were administered) data, published after the year 2015. The studies regarding metastatic and recurrent cancers as well as the prognosis of overall survival or the outcome of consecutive treatment were not taken into consideration. As revealed from the systematic inspection of the papers, there are over 100 independent parameters analyzed for their suitability as prognostic markers in HNSCC patients undergoing induction chemotherapy. Some of them are promising, but usually they lack important features such as high specificity and sensitivity, low cost, high positive predictive value, clinical relevance, short turnaround time, etc. Subsequent studies are necessary to confirm the usability of the biomarkers for personal medicine.