Head and neck squamous cell cancer (HNSCC) is one of the ten most common malignant neoplasms, characterized by an aggressive course, high recurrence rate, poor response to treatment, and low survival rate. This creates the need for a deeper understanding of the mechanisms of the pathogenesis of this cancer. The tumor microenvironment (TME) of HNSCC consists of stromal and immune cells, blood and lymphatic vessels, and extracellular matrix. It is known that HNSCC is characterized by complex relationships between cancer cells and TME components. TME components and their dynamic interactions with cancer cells enhance tumor adaptation to the environment, which provides the highly aggressive potential of HNSCC and resistance to antitumor therapy. Basic research aimed at studying the role of TME components in HNSCC carcinogenesis may serve as a key to the discovery of both new biomarkers–predictors of prognosis and targets for new antitumor drugs. This review article focuses on the role and interaction with cancer of TME components such as newly formed vessels, cancer-associated fibroblasts, and extracellular matrix.